Equivariant covering type and the number of vertices in equivariant triangulations

Abstract

We introduce the notion of the equivariant covering type of a space X on which a finite group G acts and study its properties. The equivariant covering type measures the size of G-equivariant good covers of X and is thus an extension of the covering type of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a G-homotopy invariant and describe its relation with other G-invariants, like the equivariant LS-category, G-genus, and the multiplicative structures of equivariant cohomology theories. We also compute the G-covering type of regular G-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of G and cohomology spheres. As an application, we derive estimates of sizes of minimal G-triangulations for various G-spaces.

1. Introduction

Given a topological space X with an action of a finite group G, we consider G-equivariant good covers of X and define the equivariant covering type, $\mathrm{ct}_G(X)$ of X as the minimal size of such a cover for spaces that are G-homotopy equivalent to X (see 3.10 for a precise definition). The equivariant covering type extends the ordinary covering type, $\mathrm{ct}(X)$ of X, which was introduced in 2016 by Karoubi and Weibel [20] and was later studied by the present authors in [12–14] and by Duan, Marzantowicz, and Zhao in [10]. In spite of some formal similarities between $\mathrm{ct}_G(X)$ and $\mathrm{ct}(X)$, the intricacies of group actions lead to completely new phenomena that required introduction of new techniques from equivariant homotopy theory.

G-covering type can be viewed as a homotopy invariant measure of the complexity of the action of G on X. It is closely related to other G-homotopy invariants like the G-equivariant Lusternik–Schinirelmann category and the G-genus of X. In this article, we develop the basic theory of the equivariant covering type. In particular, we derive several lower bounds for $\mathrm{ct}_G(X)$ in terms of the equivariant LS-category, G-genus, and the cup-length in various equivariant cohomology theories.

An important property of $\mathrm{ct}_G(X)$ is that it gives a lower bound for the number of orbits of vertices in any regular triangulation of X. In fact, classical theorems on the triangulability of G-actions on smooth manifolds proved in the 1970s by Illman [18, 19] and others do not provide any information about the size of those triangulations. While for general polyhedra, there exists a well-established study of minimal triangulations (mostly based on methods of combinatorial geometry—see surveys by Datta [8] and Lutz [21]), as far as we know, there does not exist an analogous theory of minimal G-triangulations. Thus, as a second main contribution of this article, we initiate the study on minimal regular G-equivariant triangulations (cf. Bredon [3, def. III 1.2]).

1.1. Main results

1. – Proposition3.12 relating the equivariant covering type and the size of regular equivariant triangulations.

2. – Theorem 3.13, in which we compute the G-covering type of a regular G-graph.

3. – Theorem 4.6, in which we give upper and lower bounds for the G-covering type of an oriented closed surface $\Sigma_\mathrm{g}$ with respect to an orientation-preserving action of a finite group G. As a consequence, we also obtain estimates of the sizes of equivariant triangulations for a regular, simplicial, and orientation-preserving action of G on $\Sigma_\mathrm{g}$. The formula connects the number of vertices of a minimal triangulation of $\Sigma^{\prime}_{\mathrm{g}^{\prime}}=\Sigma_\mathrm{g}/G$ with the number of singular fibers of a branched cover $\pi: \Sigma_\mathrm{g} \to \Sigma^\prime_{\mathrm{g}^\prime}$.

4. – Theorem 5.1, where we give a lower bound for the G-covering type $\mathrm{ct}_G(X)$ in terms of the equivariant genus $\gamma_G(X)$, which can be explicitly computed.

5. – Theorem 6.6, where we give a lower estimate of $\mathrm{ct}_G(X)$ using the weighted cohomological length of $h^*_G(X)$ for an arbitrary generalized equivariant cohomology theory $h^*_G$. As an application we obtain theorem 6.8, where we use equivariant K-theory to prove that $\mathrm{ct}_G(P(V)) \geq n^2$, where P(V) is the projectivization of a complex n-dimensional representation V of G. Another interesting application is theorem 6.17 in which we use the Borel cohomology to estimate the equivariant covering type for actions of $G = (\mathbb{Z}_p)^k$ (p a prime) on an n-dimensional $\mathbb{Z}_p$-cohomology sphere X. If the action is free, then $\mathrm{ct}_G(X)\geq \frac{(n+1)(n+2)}{2}$ for p = 2, and $\mathrm{ct}_G(X)\geq \frac{(n+1)(n+3)}{8}$ for p odd. An estimate for the case where the action has fixed points is also provided.

1.2. Outline

In §2, we introduce the notation and recall some basic definitions and results on the equivariant Lusternik–Schnirelmann category and the G-genus of G-spaces. In §3, we discuss equivariant good covers, define the equivariant covering type, and show how it is related to the G-category and G-genus. In §3.1, we compute the equivariant covering type of finite one-dimensional complexes (finite G-graphs). Section 4 is dedicated to the computation of the G-covering type for orientation-preserving finite group actions on closed oriented surfaces. In §5, we prove several lower estimates of G-covering type in terms of G-genus and G-category. Finally, §6 is dedicated to estimates of the equivariant covering type of some classes of G-spaces based on the multiplicative structure of various generalized cohomology theories.

1.3. Thanks

The final form of this work was shaped during the workshop ‘Some problems of Applied & Computational Topology’ which was held 05.03.2023–11.03.2023 in the Bedlewo Conference Center. The authors would like to express their thanks to the authorities of Banach Center for giving them an opportunity to organize this meeting.

2. Equivariant Lusternik–Schnirelman category

2.1. All groups considered in this work are finite

Nevertheless, we will occasionally include the finiteness assumption in the statements of the theorems to make them self-contained. When discussing concepts from equivariant topology and homotopy theory we will use freely the terminology and notation from Bredon [3].

The elementary (indecomposable) G-sets are the homogeneous sets of cosets $G/H$, where $H\le G$ is an arbitrary subgroup. Given subgroups $H,K\le G$ and $g\in G$, the formula $\varphi(gK):=gH$ gives a well-defined G-map $\varphi: G/K \to G/H$ between respective orbits if, and only if, $gKg^{-1} \subseteq H$. As a consequence, the set of G-maps $\textrm{Map}_G(G/K,G/H)$ can be identified with the set of invariants $(G/H)^K$.

A topological space X with a continuous action of a group G will be called a G-space. Given $x\in X$ the set $Gx=\{gx\mid g\in G\}$ is the orbit of x. Every orbit Gx can be identified with the set of cosets $G/G_x$, where $G_x=\{g\in G\mid gx=x\}$ is the stabilizer of the action of G at the point x. A subset $A\subset X$ is G-invariant if for every $g\in G$ we have gA = A. Clearly, the orbits are the minimal G-invariant subsets of X and every invariant subset of X can be partitioned as a disjoint union of orbits. More generally, for any $A\subset X$ we define the saturation $GA=\bigcup_{g\in G} gA$, which is the smallest invariant subset of X containing A. An open cover $\mathcal{U}$ of X is a G-cover if $gU\in \mathcal{U}$ for every $g\in G$ and $U\in \mathcal{U}$. Every G-cover $\mathcal{U}$ can be partitioned into orbits $\widetilde{U}=\{gU\mid g\in G\}$ which are G-invariant subfamilies of $\mathcal{U}$.

The classical Lusternik–Schnirelmann category $\mathrm{cat}(X)$ of a space X is defined as the minimal number of open categorical sets that are needed to cover X, where a subspace $U\subseteq X$ is called categorical if the inclusion $U\hookrightarrow X$ is homotopic to a constant map, see Cornea, Lupton, Oprea, and Tanre [7]. In the equivariant setting, we require maps and homotopies to be G-equivariant, and so we need to replace points with G-orbits. Thus, a G-invariant subset $U\subseteq X$ is said to be G-categorical if there is a G-equivariant homotopy between the inclusion $U\hookrightarrow X$ and the inclusion of some orbit $Gx\hookrightarrow X$. Note that a G-categorical set that deforms to an orbit Gx must have at least $|Gx|=|G/G_x|$ components which are mapped to different points of Gx. If there is an equivariant map from Gx to Gx′, then a G-homotopy to Gx can be extended to a G-homotopy to Gx′. In particular, if G has a fixed point x ₀, then $Gx_0=\{x_0\}$ and so every G-categorical set is a categorical set in the classical sense. This is why we normally consider G-actions without fixed points, or alternatively, restrict the types of orbits to which the invariant subsets of X can be deformed.

Let $\mathcal{A} $ be some subset of the set $\{G/H\mid H\le G\}$. An open G-invariant subset $U\subset X$ is said to be G-categorical in X with respect to $\mathcal{A} $ if $U\hookrightarrow X$ is G-homotopic to the inclusion $Gx\hookrightarrow X$ for some $Gx\in\mathcal{A}$. For a G-space X, we define the G-category of X with respect to $\mathcal{A}$ as the minimal cardinality (denoted $\mathrm{cat}_G^\mathcal{A}(X)$) of a cover of X by open sets that are G-categorical with respect to $\mathcal{A}$. The most common choice for $\mathcal{A}$ is the family of all non-trivial orbits $\mathcal{A} = \{G/H\mid H\neq G\}$. In that case, we will write $\mathrm{cat}_G(X)$ instead of $\mathrm{cat}_G^\mathcal{A}(X)$. Observe that if the action of G has fixed points, then $\mathrm{cat}_G(X)=\infty$, so we will often assume that the action of G is fixed point free.

The equivariant version of the Lusternik–Schnirelman category has been introduced by Fadell [11] for free G-actions and by Clapp and Puppe [5] and Marzantowicz [22] for the general case. Standard references for G-category are Bartsch [1] and Marzantowicz [22]. We will also need some specific computations of the equivariant category for finite group actions on surfaces from Gromadzki, Jezierski, and Marzantowicz [15].

There is another numerical invariant related to the G-category which is often easier to compute. First of all, note that the join $X*Y$ of two G-spaces is again a G-space with respect to the diagonal action defined as $g(x,t,y):= (g x,t,g y)$. Given a set of orbits $\mathcal{A}$, we define the G-genus of X relative to $\mathcal{A}$ (denoted $\gamma_G^\mathcal{A}(X)$) as the minimal integer n for which there exist orbits $A_1,\ldots,A_n\in\mathcal{A}$ and a G-map $X\longrightarrow A_1*\cdots * A_n$. If no such n exists we set $\gamma^\mathcal{A}_G(X)= \infty$. As before, if $\mathcal{A}$ is the set of all non-trivial orbits of G, then we abbreviate $\gamma^\mathcal{A}_G(X)$ to $\gamma_G(X)$.

The main properties of the G-genus are summarized in the following propositions.

Proposition 2.1. [1, proposition 2.10]

For any G-space X, we have $\; \gamma^\mathcal{A}_G(X)\, \leq \,cat^\mathcal{A}_G(X)\,$.

Proposition 2.2. ([1, proposition 2.15])

• • G-invariance: If X and Y are G-homotopy equivalent spaces, then $\gamma^\mathcal{A}_G(X)= \gamma^\mathcal{A}_G(Y)$.

• • Normalization: $\gamma^\mathcal{A}_G(X) = 1$ if, and only if X is G-contractible relative to $\mathcal{A}$.

• • Monotonicity: If there exists a G-map X → Y, then $\gamma^\mathcal{A}_G(X) \leq \gamma^\mathcal{A}_G(Y)$.

• • Subadditivity: $\gamma^\mathcal{A}_G(X \cup Y ) \leq \gamma^\mathcal{A}_G(X) + \gamma^\mathcal{A}_G(Y)$.

Proposition 2.3 ([1, proposition 2.16])

If X is a compact G-space, then $\gamma_G(X) \lt \infty$. If, in addition, there exists an orbit type $G/H\in\mathcal{A}$, such that every orbit $Gx\subset X$ can be G-equivariantly mapped to $G/H$, then

\begin{equation*}\gamma^\mathcal{A}_G(X)\leq \dim X +1.\end{equation*}

The latter assumption is satisfied if there exists one minimal orbit type in $\mathcal{A}$.

The G-category shares only the G-homotopy invariance, normalization, and subadditivity from the above listed properties. The estimate of proposition 2.3 takes a much more complicated form in the case of G-category (cf. [1, 6, 22]), and the upper bound is greater than $\dim X +1$ in general.

We conclude the section with a short proof that G-category is invariant with respect to G-homotopy equivalences, as this is not mentioned in the cited references.

Definition 2.4. We say that a G-map $\phi: X\to Y$ preserves the family of orbits $\mathcal{A}= \{(G/H_s)\}$ if for every $X \supset Gx \simeq G/G_x\in \mathcal{A}$, we have $\phi(Gx) \simeq G/ G_{\phi(x)} \in \mathcal{A}$.

Note that the condition of definition 2.4 is satisfied if $X^G=Y^G=\emptyset$ and $\mathcal{A}=\{G/H\mid H\neq G\}$.

Proposition 2.5. Let X, Y be G-spaces. If a G-homotopy equivalence between X and Y can be given by G-maps $\phi\colon X\to Y$ and $\psi\colon Y\to X$ that preserve $\mathcal{A}$, then $\mathrm{cat}_G^\mathcal{A}(X)=\mathrm{cat}_G^\mathcal{A}(Y)$.

Proof. Let $\phi: {X\overset{G}{\,\to\,}} Y$ and correspondingly $\psi: Y{\overset{G}{\,\to\,}} X$ be such a G- maps that $\psi \circ \phi {\overset{G}{\,\sim\,}} \textrm{id}_X$ and $\phi \circ \psi {\overset{G}{\,\sim\,}} \textrm{id}_Y$ respectively. Furthermore, let $\tilde{V}_1, \, \dots, \tilde{V}_k$, $k=\mathrm{cat}_G^\mathcal{A}(Y)$ be a G-categorical cover of Y, and $r_{i,t}: \tilde{U}_i \times I \to Y$, $1\leq i \leq k$, with $ r_{i,1}: \tilde{U}_i \to G/G_{y_i}$, the corresponding G-maps. Then the G-cover $\tilde{U}_i:= \phi^{-1}(V)$ of X is G-categorical. Indeed, the composition $ \psi \circ r_{i,t} \circ \phi : \tilde{U}_i \times I \to X $ gives the required G-deformation, because $\psi \circ r_{i,1} \circ \phi : \tilde{U}_i \to \psi(G/ G_{y_i}) \subset X$, with $\psi(G/G_{y_i})\in \mathcal{A}$, and $\psi \circ\ r_{i,0} \circ \phi : \tilde{U}_i \to X$ is G-homotopic to $\psi\circ\phi\colon \tilde{U}_i\to X$. But the latter is G-homotopic to the inclusion of $\tilde{U}_i$ in X, which shows $\mathrm{cat}_G^\mathcal{A}(X)\leq \mathrm{cat}_G^\mathcal{A}(Y)$. The proof of the converse inequality is analogous.

3. Equivariant covering type

The covering type of a space is a homotopy invariant that was recently introduced by Karoubi and Weibel [20]. It is based on open covers that are good in the sense that all finite, non-empty intersections of elements of the cover are contractible. Then the covering type $\mathrm{ct}(X)$ of X is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to X. The well-known nerve theorem (see Hatcher [16, cor. 4G.3]) states that for every good cover $\mathcal{U}$ of a paracompact space X the geometric realization $|N(\mathcal{U})|$ of the nerve of the cover is homotopy equivalent to X (in other words, $N(\mathcal{U})$ is a homotopy triangulation of X). Thus, we can say that the covering type of a paracompact space X equals the minimal number of vertices in a homotopy triangulation of X.

In order to extend the above results to the equivariant context, we need some preparation. A G-space X is said to be G-contractible, if there exists an $x\in X$, such that the orbit $Gx\subset X$ is a G-deformation retract of X.

The proof of the following proposition is straightforward.

Proposition 3.1. Let X be G-contractible to an orbit Gx. Then X has precisely $|Gx|$ path-components and each of them is contractible to a point in the orbit Gx.

Given a G-space X, we are interested in triangulations of X that are compatible with the group action. Recall that an abstract simplicial complex is determined by a set of vertices V and a set K of finite non-empty subsets of V, called simplices, which is closed under inclusion (i.e. a non-empty subset of a simplex in K is also a simplex in K). A simplicial map between two simplicial complexes is a map between the respective sets of vertices that carries simplices into simplices.

Definition 3.2. (cf. Bredon [3, Sec. III.1])

1. (1) A simplicial G-complex is a simplicial complex K together with an action of G on K by simplicial maps.

2. (2) A simplicial G-complex K is regular if the action of G on K satisfies the following conditions:

   1. (R1) If vertices v and gv belong to the same simplex in K, then v = gv.

   2. (R2) If $\langle v_0,\ldots,v_n\rangle$ is a simplex of K and if for some choice of $g_0,\ldots,g_n\in G$ the points $g_0 v_0,\ldots , g_n v_n$ also span a simplex of K, then there exist $g\in G$, such that $gv_i= g_i v_i$, for $i =0,\ldots\, n$ (in other words, $\langle g_0 v_0,\ldots , g_n v_n\rangle=g\langle v_0,\ldots,v_n\rangle$).

We will also need a slightly stronger regularity condition, introduced by Illman.

Definition 3.3. (cf. Illman [18, p. 201]) A simplicial G-complex K is an equivariant simplicial complex or a strictly regular G-complex if it satisfies (R1) and the following condition:

1. (R3) For any n-simplex σ of K, the vertices $v_0, \,\dots, \, v_n$ of K can be ordered in such a way that we have $G_{v_n} \subset G_{v_{n-1}} \subset \, \cdots\, \subset G_{v_0}$.

By conditions (R2) or (R3), if two n-simplices in K have vertices from the same set of orbits, then they belong to an orbit of the action of G on K. Thus, if K is a regular G-complex, then one can naturally build a quotient simplicial complex $K/G$ whose vertices are the orbits of the action of G on the vertices of K, and whose simplices are the orbits of the action of G on the simplices of K. Clearly, the geometric realization $|K/G|$ of the quotient complex is homeomorphic to the quotient space $|K|/G$.

The regularity conditions are quite stringent. For example, none of them hold for the ${\mathbb{Z}}_3$-action that rotates the 2-simplex around its centre. The following proposition shows that the situation improves if we consider barycentric subdivisions.

Proposition. ([3, prop. III.1.1]) If K is any simplicial G-complex, then the induced action on the barycentric subdivision K ^′ satisifies condition (R1). Moreover, if the action of G on K satisfies R1), then the induced action on K ^′ satisfies (R2) and (R3).

Therefore, any simplicial action of G on K induces a strictly regular action, on the second barycentric subdivision of K.

Thus, we can always achieve regularity of a group action by increasing the number of simplices, but here our goal is to estimate what is the minimal size of a triangulation for which a given group action is regular. To this end, we need a relation between simplicial G-complexes and G-covers. The following notions were introduced by Yang [27].

Definition 3.5. An open G-cover $\mathcal{U}$ of a G-space X is regular if the following conditions hold:

1. (RC1) For every $U \in \mathcal{U}$ and $g\in G$, either U = gU or $U\cap gU=\emptyset$.

2. (RC2) If $U_0,\ldots, U_n$ are elements of $\mathcal{U}$ with non-empty intersection and if for some choice of elements $g_0,\ldots,g_n\in G$ the intersection of sets $g_0U_0,\ldots, g_nU_n$ is also non-empty, then there exists $g\in g$ such that $gU_i=g_iU_i$ for $i \leq n$.

In short, $\mathcal{U}$ is a regular G-cover if its nerve $N(\mathcal{U})$ is a regular G-complex.

Let $\mathcal{U} =\{U_\alpha\}_{\alpha \in I} $ be an open G-cover of G-space X. For any subgroup $H \subset G$ and $\alpha \in I$, let $U^H_\alpha = U_\alpha \cap X^H$. Denote by $\mathcal{U}^H$ the collection of $\{U^H_\alpha\}_{\alpha \in I}$. It is clear that $\mathcal{U}^H$ is an open cover of X^H. The following definition is a natural extension of the concept of a good cover to the equivariant setting.

Definition 3.6. A regular open G-cover $\mathcal{U}$ (split into orbits $\tilde{U}= GU$) is said to be a good G-cover if all orbits $\tilde{U}$ of elements of $\mathcal{U}$ and all their non-empty finite intersections are G-contractible.

Remark 3.7. It follows immediately from the definition of a good G-cover $\mathcal{U}$ that the set of images $\mathcal{U}^*:=\{\pi(U)\mid U\in\mathcal{U}\}$ with respect to the projection $\pi\colon X\to X/G$ forms a good cover of the orbit space $X^*=X/G$.

Yang [27] introduced another definition of good G-covers which may look less natural but is more convenient if one wants to state and prove an equivariant version of the nerve theorem. In the next proposition, we show that the two approaches are in fact equivalent.

Proposition 3.8. A G-cover $\mathcal{U} =\{U_s\}$ (split into orbits $\tilde{U}_{i\in I}$) is a good G-cover of a G-manifold X if, and only if it is a regular G-cover and $\mathcal{U}^H$ is a good cover of X^H for all subgroups $ H\subset G$.

Proof. $\mathcal{U} =\{U_s\}$, split into orbits $\tilde{U}_{i\in I}$ be a good G-cover of X in the sense of definition 3.6, and $r: \tilde{U}_{i_1} \cap \, \cdots\, \cap \tilde{U}_{i_k} \to Gx$ a G-deformation retraction. Then the restriction r^H of r to $(\tilde{U}_{i_1} \cap \, \cdots\, \cap \tilde{U}_{i_k})^H$ is a deformation retract of each intersection of $(U_{i_1} \cap \, \cdots\, \cap U_{i_k})^H$ onto the unique point $x\in (Gx)^H $ which is in $U_{i_1} \cap \, \cdots\, \cap U_{i_k}$. This shows that all sets $(U_{i_1} \cap \, \cdots\, \cap U_{i_k})^H$ are contractible which shows that this G-cover satisfies the stated condition.

To show the converse implication, we use an adaptation of the argument of proof of corollary [27, 2.12]. Let $\mathcal{U} =\{U_{s\in S}\}$ split into orbits $\{\tilde{U}_{i\in I}\}$, $ \tilde{U}_i = {\underset{g \in G}\cup}\, g U_{s}$, be a G-cover of X such that $ \mathcal{U}^H :=\{U^H_{s\in S}\}$ is a good cover X^H for all $H\le G$. Since $\mathcal{U} =\{U_{s\in S}\}$ is regular G-cover, $g\,U_s \cap g^\prime\,U_s \neq \emptyset$ implies $g\,U_s = g^\prime\,U_s $. A non-empty intersection

\begin{equation*} \tilde{W} = \tilde{U}_{i_1} \cap \, \cdots \,\cap \, \tilde{U}_{i_n}\end{equation*}

is of the form

(3.1)\begin{equation} {\underset{i_1, \, \dots\, i_n }{\, \bigcup\,}}\; g_{i_1} U_{i_1}\cap \,\cdots\,\cap \, g_{i_n} U_{i_n}. \end{equation}

If the isotropy group $G_{g_{i_k} U_{i_k}}= H_{i_k}$ then the isotropy group of $ g_{i_1} U_{i_1}\cap \,\cdots\,\cap \, g_{i_n} U_{i_n}$ is $ H= {\underset{1}{\overset{n}\cap}}\, H_{i_k}$, i.e. $ g_{i_1} U_{i_1}\cap \,\cdots\,\cap \, g_{i_n} U_{i_n}= ( g_{i_1} U_{i_1}\cap \,\cdots\,\cap \, g_{i_n} U_{i_n})^H$. Since every $U_{i_k}$, thus $ g_{i_k} U_{i_k}$, is contractible, every non-empty summand of $\tilde{W}$ in (3.1) is contractible by our assumption applied for H.

Since $\tilde{W}$ is non-empty, there exists non-empty contractible space $D= g_{i_1} U_{i_1}\cap \,\cdots\, \cap g_{i_n} U_{i_n}$. If $D^\prime = g^\prime_{i_1} U_{i_1}\cap \,\cdots\, \cap g^\prime_{i_n} U_{i_n} $ is also non-empty, then the condition (RC2) of definition 3.5 implies there exists $h\in $G such that $g_k^\prime U_{i_k} = h(g_k U_{i_k})$ for $k= 1, 2,\, \dots, n$. Hence $D^\prime = hD$.

Since $\tilde{W} $ is G-invariant, W has the form $G/H \times D$, where D is contractible and the action of G is on the first factor.

Let $r_t: D\times [0,1] \to D$ be the deformation of D to a point $[x]= r_1(D)$, where we identify D with its image $\pi(D)\subset X/G$. Since over $D\subset X/G$ there is only one orbit type, the deformation $r_t: D\times [0,1] \to D$ lifts to a G-deformation $\tilde{r}_t: \tilde{U}_{i_1} \cap \,\cdots\, \cap \tilde{U}_{i_k} \times [0,1] \to \tilde{U}_{i_1} \cap \,\cdots\, \cap \tilde{U}_{i_k}$. This shows that this intersection is G-contractible to an orbit $ Gx\simeq G/H =r_1(\tilde{U}_{i_1} \cap \,\cdots\, \cap \tilde{U}_{i_k})$. Consequently, $\mathcal{U}$ satisfies the condition of definition 3.6.

The above implication can be also shown by another argument. For a G-cover of X $\mathcal{U} =\{U_{s\in S}\}$, split into orbits $\{\tilde{U}_{i\in I}\}$, $ \tilde{U}_i = {\underset{g \in G}\cup}\, g U_{s}$, such that $ \mathcal{U}^H :=\{U^H_{s\in S}\}$ is a good cover of X^H for all $H\le G$ we take the geometric realization of its nerve $\mathcal{N}(\mathcal{U})$. Under this assumption, the Alexandrov map $\phi: X \to |\mathcal{N}(\mathcal{U})|$ is G-homotopy equivalence as follows from theorem 3.9. Then the G-cover $\tilde{W}$ of $|\mathcal{N}(\mathcal{U})|$, consisted of saturations of the stars of vertices of the nerve $\mathcal{N}(\mathcal{U})$, satisfies the condition of definition 3.6 as follows from corollary [27, 2.12] and the above argument. Consequently, $\tilde{\mathcal{U}}^\prime = \phi^{-1} (\tilde{W})$ gives a regular good G-cover of X of the same cardinality as $\tilde{U}$ which satisfies definition 3.6.

Theorem 2.11 of Yang [27] states that every smooth G-manifold has a good G-cover. The equivariant good covers are in fact co-final in the set of open covers of a G-manifold X. Moreover, Yang [27, theorem 2.19] proved the following equivariant version of the nerve theorem.

Theorem 3.9. If $\mathcal{U}$ is a locally finite, e.g. finite, equivariant good cover of a G-CW complex X, then the usual realization $|\mathcal{N}(\mathcal{U})|$ of the nerve $\mathcal{N}(\mathcal{U})$ is G-homotopy equivalent to X.

We are now prepared to define the main concept of this article.

Definition 3.10. The strict G-covering type of a given G-space X is the minimal cardinality $\mathrm{sct}_G(X)$ of the set of orbits of a G-invariant regular good cover for X.

Furthermore, the G-covering type of a G-space X is the minimal value of $\mathrm{sct}_G(Y)$ among spaces Y that are G-homotopy equivalent to X, i.e.

\begin{equation*}\mathrm{ct}_G(X):=\min\{\mathrm{sct}_G(Y)\,\mid\, Y {\overset{G}{\,\simeq\,}} X\}.\end{equation*}

Note that $\mathrm{sct}_G(X)$ and $\mathrm{ct}_G(X)$ can be infinite (e.g. if X is an infinite discrete space) or even undefined, if the space does not admit any good covers. In what follows we will always tacitly assume that the spaces under consideration admit finite good covers.

It is clear from the definitions that a G-invariant regular open cover $\mathcal{U}$ of X induces an open good cover of the orbit space $X/G$ as the projection map $\pi: X \to X/G$ is open and G-contraction of $\tilde{U}$ to an orbit Gx induces a contraction of $p(\tilde{U})$ to $*=[Gx]$ in $X/G$. Thus, we get the following result.

Corollary 3.11. For a G-space X which is a G-CW complex we have

\begin{equation*} \mathrm{sct}(X/G) \leq \mathrm{sct}_G(X) \;\; \; {\text{and }} \;\;\; \mathrm{ct}(X/G) \leq \mathrm{ct}_G(X).\end{equation*}

In the proof of proposition 3.8, we used the fact that open stars of a regular G-complex K are a good G-cover of $|K|$. In view of definition 3.6, this immediately yields the following result.

Proposition 3.12. Let K be a simplicial G-complex with a regular action of G. Then

\begin{equation*} \mathrm{ct}_G(|K|)\leq \mathrm{sct}_G(|K|) \leq \Delta^*(K),\end{equation*}

where $\Delta^*(K)$ denotes the number of orbits of vertices of K.

In discrete geometry, to every finite simplicial complex K of dimension d, one can associate the so called f-vector $\vec{f}(K)=(f_0(K), f_1(K), \,\dots\, , f_d(K))$, where $f_i(K)$ is the number of i-dimensional faces of K. Analogously, given a d-dimensional simplicial G-complex K, we define the equivariant f-vector

(3.2)\begin{equation} \vec{f}_G(K):= (f_{G,0}(K), f_{G,1}(K), \dots , f_{G,d}(K)) \;\; \end{equation}

where $f_{G,i}(K)$ is the number of orbits of i-dimensional simplices of K.

Note that the f-vector and the equivariant f-vector of a simplicial G-complex are related by the formula

\begin{equation*}f_i(K)= {\underset{\sigma_i}\sum} \, |G/G_{\sigma_i}| = {\underset{1}{\overset{f_{G,i}}{\,\sum}}}\,|G/G_{\sigma}|,\end{equation*}

where the sum is taken over representatives of all orbits of i-simplices σ of K or equivalently of all i-simplices of the induced triangulation of $K/G$.

The main aim of this article is to give some lower estimates of $f_{G,0}(K)$ and $f_{0}(K)$.

3.1. G-covering type of one-dimensional complexes

Our first explicit example is the computation of the G-covering type of a finite G-graph, i.e. a space $X=|K|$, where K is a one-dimensional simplicial complex with a regular simplicial action of a group G. In other words, G permutes vertices and edges of K and for every edge $e=[v_1,v_2]$, if $g[v_1, v_2] \subset [v_1,v_2]$, then $g=\textrm{id}_{[v_1,v_2]}$.

Before proceeding, let us recall the corresponding result for the non-equivariant case (cf. Karoubi–Weibel [20, proposition 4.1]). If X_h is a bouquet of h circles then

(3.3)\begin{equation}\mathrm{ct}(X_h) =\Bigg\lceil\,\frac{3+\sqrt{1+8h}}{2}\,\Bigg\rceil \end{equation}

where $ \lceil \alpha \rceil$ denotes the ceiling of a real number α. In other words, $\mathrm{ct}(X_h)$ is the unique integer n such that

(3.4)\begin{equation} \binom{n-2}{2} \lt h \leq \binom{n-1}{2}. \end{equation}

Let X be a finite G-graph and let $\mathcal{U}$ be a regular G-cover of X. As before, denote by $\tilde{\mathcal{U}}^{*}$ the induced cover of $X/G$. Suppose first that $X=X_{(H)}$, i.e. we have one orbit type (H). In this case, we have $\mathrm{ct}_G(X)= \mathrm{ct}(X/G)$, because every deformation of $\tilde{U}_{i_1}^*\cap\cdots\cap \tilde{U}^*_{i_k}$ of projections $\tilde{U}^*_{i_j}= \pi(\tilde{U}_{i_j})$ to a point $[x]=\pi(x)\in X^*=X/G$ in $\tilde{U}_{i_1}^*\cap\cdots\cap\tilde{U}^*_{i_k}$ can be lifted to a G-deformation of $\tilde{U}_{i_1}\cap\cdots\cap\tilde{U}_{i_k}$ to the orbit $Gx = G/H$. Observe that topological loops in $X^*$ come from two different types of situations:

1. (i) if $|G/H|= m$ and G acts as the cyclic group of order m on a cycle of the form

   \begin{equation*}[v_1, v_2],[v_2,v_3],\ldots,[v_m,v_1],\end{equation*}

   with the action $v_i\mapsto v_{i+1} \mod m$.

2. (ii) if the loop is given as the graph cycle $[w_1, w_2], [w_2,w_3],\dots , [w_n,w_1]$ and $gw_i\notin \{w_1, \dots , w_n\}$ for any $g\in G$. Consequently, for every $g\in H$ such that $[g]\neq [H] \in G/H$ the image of the cycle $[w_1, w_2], [w_2,w_3], \dots , [w_n,w_1]$ is another disjoint cycle $[w^\prime_1, w^\prime_2], [w^\prime_2,w^\prime_3] \dots , [w^\prime_n, w^\prime_1]$. The group G is acting trivially on elements of each such cycle and is permuting these cycles, i.e. each cycle can be identified with an element of $G/H$ and G is permuting them transitively as it permutes elements of $G/H$.

Consequently, every one-dimensional finite regular G-graph is, up to G-homotopy, a union of topological loops of type (i) or (ii). Assume there are k cycles of type i), and $l \times |G/H|$ cycles (with the corresponding action) of type (ii). This shows that $\mathrm{ct}_G(X)=\mathrm{ct}(X^*)= k +l$, i.e. the number of loops in the bouquet X_h of $h=k+l$ circles being homotopy equivalent to $X^*=X/G$. Applying (3.3), as a result we get a formula for G-covering type of a finite graph X such that $X=X_{(H)}$

(3.5)\begin{equation} \mathrm{ct}_G(X)= \bigg\lceil\,\frac{3+\sqrt{1+8(k+l)}}{2}\,\bigg\rceil, \end{equation}

with X having k cycles of type (i) and l orbits of cycles of type (ii).

We handle the general case by using induction with respect to the partially ordered set of all orbit types $\mathcal{S}_G$ with the order $ (H) \succ (K) \, \Longleftrightarrow \, K \subset gHg^{-1}$ for some $g\in G$. For a given G-space X, by $\mathcal{S}_G(X)$, we denote the subset of $\mathcal{S}_G$ consisting of (H) for which $X_{(H)}\neq\emptyset$. Obviously, we can restrict our consideration to $(H)\in \mathcal{S}_G(X)$.

Let $N_\epsilon(A)$ denote open and invariant neighbourhood of invariant set A. If X is a G-graph and $A {\overset{G}\subset} X$ is an invariant subcomplex then as $N_\epsilon(A)$ we can take the union of open stars of all vertices of A, i.e. the union of all edges of A and half-open edges of length ϵ of X which have one vertex in A. In the case of G-graphs, we can take always ϵ = 1, i.e. the open star $N_1(A)=\textrm{st}(A)$.

In the inductive step, suppose that $X=X^{(H)}= X_{(H)}\sqcup X^{(K)\nsucc (H)}$. In general, there exists ϵ > 0 such that $X^{(K)\nsucc (H)}$ is a G-deformation retract of the open and invariant neighbourhood $N_\epsilon(X^{(K)\nsucc (H)})$, but here it holds for $N_1(X^{(K)\nsucc (H)})$.

Let $X^\prime_{(H)}$ be the compact closed set (graph) $X\setminus N_\epsilon(X^{(K)\nsucc (H)})$, and let $0\leq h_{(H)}(X)$, or shortly $h_{(H)}$, be the number of loops of $ X^\prime_{(H)}/G$, i.e. the number of generators of $\pi_1(X^\prime_{(H)}/G)$.

We can now state the main result of this subsection.

Theorem 3.13. Let X be a finite connected graph with a regular simplicial action of a group G, and let $\mathcal{S}_G(X):=\{(H)\in \mathcal{S}_G\mid X_{(H)}\neq \emptyset \}$. Then

\begin{equation*} \mathrm{ct}_G(X)\, =\, \sum_{(H)\in \mathcal{S}_G(X)} \, \Bigg\lceil\,\frac{3+\sqrt{1+8h_{(H)}}}{2}\,\Bigg\rceil \,. \end{equation*}

Proof. Let (H) be a minimal orbit type in $\mathcal{S}_G(X)$. For $X=X^{(H)}= X_{(H)}$, we have already shown the statement, cf. (3.5). Suppose that $X=X^{(H^\prime)}= X_{(H^\prime)} \sqcup X_{(H)}$, with $X_{(H)}=X^{(H)}$, where $(H^\prime)\succ (H)$ and that there does not exists $(K)\neq (H^\prime), (H)$ such that $(H^\prime)\succ (K)\succ (H)$.

Let $\mathcal{U}=\{\tilde{U}_i\}$, $1\leq i \leq \mathrm{ct}_G(X)$, be a regular good G-cover of X consisting of G-contractible saturations of open stars of vertices. Decompose $\mathcal{U} = \mathcal{U}_1 \sqcup \mathcal{U}_2$ into two subfamilies of cardinality $c_1,\, c_2$, respectively, where $\mathcal{U}_1$ consists of $\tilde{U}_i$ for which $\tilde{U}_i \cap X^{(H)} \neq \emptyset$, and $\mathcal{U}_2$ consists of $\tilde{U}_i$ for which $\tilde{U}_i \cap X^{(H)} =\emptyset$. Note that if $\tilde{U}_i \cap X^{(H)} \neq \emptyset$ then $\tilde{U}_i $ is G-contractible to an orbit Gx, where x is a vertex in $X^{(H)}$, because the contraction is a G-map. Moreover, ${\underset{j=1}{\overset{c_2} {\,\cup\,}}} \tilde{U}_j = X_{(H^\prime)}$, because $X_{(H^\prime)} = X^{(H^\prime)}\setminus X^{(H)}$ and $X^{(H)}$ consists of all edges, such that both vertices are in $X^{(H)}$.

By proposition 3.8, sets $\{\tilde{U}_i\}_{i=1}^{c_1}$ form a regular good G-cover of $ X^{(H)}$ thus $c_1 \geq \mathrm{ct}_G(X^{(H)})$. Suppose that $c_1 \lt \mathrm{ct}_G(X_{(H)}) = \mathrm{ct}_G(X^{(H)})$. Then there a exists a good G-cover $\{\tilde{U}_i^\prime\}_{i=1}^{c^\prime_1}$ of $ X^{(H)})$ with $ c^\prime_1 \lt c_1$. Let $r: N_1(X^{(H)}) \to X^{(H)}$ be a G-deformation retraction. Then $\{\tilde{U}_i^{\prime\prime}\}_{i=1}^{c^\prime_1}= r^{-1}(\{\tilde{U}_i^\prime\}_{i=1}^{c^\prime_1}$ is a good G-cover of $N_1(X^{(H)})$, because we can compose the G-deformation retraction $r: N_\epsilon(X^{(H)})\to X^{(H)}$ with corresponding G-contractions of $\{\tilde{U}_i^\prime\}$. Since ${\underset{i=1}{\overset{c_1^\prime} {\,\cup\,}}} \tilde{U}_i^{\prime\prime}\cup {\underset{j=1}{ \overset{c_2} {\,\cup\,}}} \tilde{U}_j = N_\epsilon(X^{(H)}) \cup X_{(H_\prime)} = X$, it gives a good G- cover of X of cardinality $c_1^\prime + c_2 \lt c_1+ c_2= \mathrm{ct}_G(X)$ which leads to a contradiction.

Now we have to show that $c_2= \mathrm{ct}_G(X_{(H^\prime)})$. Since $\{\tilde{U}_j\}_{j=1}^{c_2}$ form a good G- cover of $X_{(H^\prime)}$ we have $c_2 \geq \mathrm{ct}_G(X_{(H^\prime)})$. If $c_2 \gt \mathrm{ct}_G(X_{(H^\prime)})$ then we can take a good G-cover $\{\tilde{U}_j^\prime\}_{j=1}^{c^\prime_2}$ with $c_2 \gt c_2^\prime = \mathrm{ct}_G(X_{(H^\prime)})$. The union of elements $\{\tilde{U}_i\}_{i=1}^{c_1}$ and $\{\tilde{U}_j^\prime\}_{j=1}^{c^\prime_2}$ gives a good G-cover of $X^{(H)}\sqcup X_{(H^\prime)} = X$ of cardinality $c_1+c_2^\prime \lt \mathrm{ct}_G(X)$ which leads to a contradiction. Consequently $c_2= \mathrm{ct}_G(X_{(H^\prime)})$. This shows the statement for X having two different orbit types which are comparable.

If in X there are more than one minimal orbit types $(H_1), \dots , (H_r)$ and another orbit type $(H^\prime)\succ (H_i)$ for all $1\leq i \leq r$ then the same argument shows the statement for $X=X^{(H^\prime)}$. Notify that if X consists of only different minimal orbit types (not comparable) then it is not connected.

By induction over orbit types assume that the statement holds for G-graphs Y having at most n different orbit types $(H_1), \, \dots , \, (H_n)$ and $X=Y \sqcup X_{(H_{n+1})}$. By the above remark and connectedness of X, we can assume that $(H_{n+1})$ is not minimal. By repeating the argument of the first inductive step, we get $\mathrm{ct}_G(X) = \mathrm{ct}_G(Y) + \mathrm{ct}_G(X_{(H_{n+1})})$. Using the induction assumption, we get

\begin{equation*} \mathrm{ct}_G(X) = \mathrm{ct}_G(Y) + \mathrm{ct}_G(X_{(H_{n+1})} ) = \sum_{i=1}^n \, \Bigg\lceil\,\frac{3+\sqrt{1+8h_{(H_i)}}}{2} \,\Bigg\rceil + \Bigg\lceil\,\frac{3+\sqrt{1+8h_{(H_{n+1})}}}{2}\,\Bigg\rceil \,\end{equation*}

as claimed.

4. Equivariant covering type and minimal triangulations of surfaces

In this section, we determine the equivariant covering type of a closed orientable surface $\Sigma_\mathrm{g}$ of genus $\mathrm{g}$ with respect to an orientation-preserving actions of a finite group G. As a consequence, we are able to estimate the minimal number of vertices and G-orbits of vertices in a triangulation of $\Sigma_\mathrm{g}$ by a regular simplicial G-complex.

4.1. Orientation preserving actions on orientable surfaces

Let $\Sigma_\mathrm{g}$ be oriented surface of genus $\mathrm{g} \geq 0$. Suppose that G acts effectively on $\Sigma_\mathrm{g}$ preserving orientation, i.e. it is a (finite) subgroup of $\textrm{Homeo}_+(\Sigma_\mathrm{g})$. It is known (Hurwitz for $\Sigma_\mathrm{g}$ with $\mathrm{g}\geq 2$, Brouwer, Kerekjarto, and Eilenberg for $\Sigma_\mathrm{g}=S^2$ and folklore for $\Sigma_\mathrm{g}=\mathbb{T}^2$) that there exists a holomorphic structure $\mathcal{H}$ on $\Sigma_\mathrm{g}$ in which $\textrm{Homeo}_+(\Sigma_\mathrm{g})$ can be viewed as a subgroup of biholomorphic isomorphisms $\textrm{Hol}(\Sigma_\mathrm{g}, \mathcal{H})$ of $(\Sigma, \mathcal{H})$. More precisely, we have the following theorem.

Theorem 4.1 (Geometrization of action)

Given a finite group G of orientation-preserving homeomorphisms of a compact orientable surface X of genus $\textrm{g}$, there is a complex structure on X with respect to which G is a group of its conformal maps. Furthermore, the orbit space $X'=X/G$ is a compact surface of genus $\mathrm{g}' \leq \mathrm{g}$ and the relation between $\mathrm{g}$ and $\mathrm{g}'$ is given by the Riemann–Hurwitz formula (see formulas (4.1) and (4.2)).

Moreover, Hurwitz’ theorem says that the for $\mathrm{g} \geq 2$ the order of $\textrm{Hol}(\Sigma_\mathrm{g}, \mathcal{H})$ is $\leq 84(\mathrm{g} -1)$.

Let $\Sigma_\mathrm{g}$ be a compact Riemann surface of genus $\textrm{g}\geq 0$ and let G be a group of holomorphic automorphisms of $\Sigma_\mathrm{g} $. Let $ \Sigma_{\mathrm{g}\prime} = \Sigma_\mathrm{g}/G$ be the quotient surface of genus $\mathrm{g}^\prime$ with the projection $\pi:X \to X^\prime $ and let $\{x^\prime_1,\, \ldots , x^\prime_r\}$ be the set of all points over which π is branched, i.e. the image of the singular orbits. For any $x_{i,j}\in \pi^{-1}(x^\prime_j)$ being a point in the orbit over $x^\prime_j$, the isotropy group $G_{x_{i,j}}$ is cyclic of order m_j (observe that the isotropy groups of points in one orbit are conjugate). We denote by m the order of G. Now, it follows that the orbit of $x_{i,j}$ has order $n_j={m}/{m_j}$. If $x^\prime\in \Sigma_{g^\prime}\setminus \{x^\prime_1, \ldots , x^\prime_r\}$, then $\pi^{-1}(x^\prime)$ is an orbit isomorphic to G and consequently, for every $x\in \pi^{-1}(x^\prime)$, $m_x=1$ and $n_x={m}/{m_x} = m$.

Denote by $\mathcal{S}$ the set of images of singular orbits $\{x_1^\prime, \dots, x_r^\prime\} $ in $\Sigma^\prime$. We have the classical Riemann–Hurwitz formula

(4.1)\begin{equation} \mathrm{g}= 1 + m(\mathrm{g}^\prime-1) + \frac{1}{2}\, m\, {\underset{j=1}{\overset{r}\sum}}\, \left(1 -\frac{1}{m_j}\right), \end{equation}

which let us also express $\mathrm{g}^\prime$ as a function of $\mathrm{g}$

(4.2)\begin{equation} \mathrm{g}^{\prime} = 1 + \frac{1}{m}(\mathrm{g}-1) - \frac{1}{2}\,\left( {\underset{j=1}{\overset{r}\sum}}\, (1 -\frac{1}{m_j})\right)\,. \end{equation}

Riemann–Hurwitz formula gives relations between $\mathrm{g},\, \mathrm{g}^\prime\,,r, \, m\,,\{m_1\,,m_2\,,\dots\,,m_r\}$, where $\mathrm{g}\geq 0$, $m=|G|$, $2\mathrm{g} +2 \geq r\geq 0$, $m_j\mid m$, i.e. some necessary condition on the action of G on $\Sigma_\mathrm{g}$. The system $ (\mathrm{g}^\prime\,, r,\, \{m_1\,,m_2\,, \, \dots\,, m_r\}) $ is called the generating vector, provided a condition is satisfied (cf. [4, definition 2.2]). The following classical result provides a converse to the Riemann–Hurwitz formula (see [4, proposition 2.1] for more information).

Proposition 4.2. (Riemann’s Existence Theorem)

The group G acts on the surface $\Sigma_\mathrm{g}$, of genus $\mathrm{g}$, with branching data $(\mathrm{g}^\prime, r, m_1, \dots , m_r,)$ if and only if the Riemann–Hurwitz equation (4.1) is satisfied, and G has a generating $(\mathrm{g}^\prime\,, r\,, m_1,\, \dots\,, m_r)$-vector.

4.2. Minimal triangulations of orientation preserving actions on surfaces

We take as a starting point the famous formula of Jungermann and Ringel that gives the minimal number of vertices in a triangulation of a closed surface in terms of its Euler characteristic.

Theorem 4.3 (Jungerman and Ringel)

Let S be a closed surface different from the orientable surface of genus 2 (M2), the Klein bottle (N2), and the non-orientable surface of genus 3 (N3). There exists a triangulation of S with n vertices if, and only if

\begin{equation*} n\ge\left\lceil \frac{7 + \sqrt{49- 24\,\chi(S)}}{2} \right\rceil\,. \end{equation*}

If S is any of the exceptional cases (M2), (N2), or (N3), then the right-hand side of the formula should be replaced by $ \left\lceil\frac{7 + \sqrt{49- 24\chi(S)}}{2} \right\rceil +1$.

For orientable surfaces, we have $\chi(\Sigma_g)=2-2g$, so we let $n_g:=\left\lceil \frac{7 + \sqrt{1+48 g}}{2} \right\rceil$ for $g\ne 2$ and $n_2:=10$. Our goal is to obtain a similar formula that estimates the minimal number of orbits of vertices of a triangulation of $\Sigma_\mathrm{g}$ which admits a simplicial, regular, and orientation-preserving action of a group G. The basic idea is quite simple. Let $\Sigma_{\mathrm{g}^\prime}$ be the orbit space $\Sigma_\mathrm{g}/G$ and let K ^′ be its minimal triangulation. If the action of G is free, then the projection $\pi: \Sigma_\mathrm{g} \to \Sigma_{\mathrm{g}^\prime}= \Sigma_\mathrm{g}/G$ is a regular covering, so it is enough to lift a minimal triangulation K ^′ of $ \Sigma_{\mathrm{g}^\prime}$. If the action G is not free, then in order to lift K ^′ to a regular invariant triangulation K of $\Sigma_g$ we have to check a compatibility condition given by the Riemann–Hurwitz formula.

Let H be a non-trivial subgroup of G. Then the action of H has a non-trivial fixed point set $ (\Sigma_\mathrm{g})^H$ if, and only if H is a cyclic. Moreover, the set${\Sigma_\mathrm{g}^{(H)}}$ is a union of finitely many isolated orbits; therefore, ${\Sigma^{(H)}}^*$ consists of a finite number of points. Since the action is regular, ${\Sigma^{(H)}}^*$ is a subcomplex of any lift K of K ^′. Consequently, the first condition of compatibility is satisfied if every $x^\prime \in \mathcal{S}$ is a vertex of the triangulation K ^′, i.e. $\mathcal{S} \subset (K^\prime)^{(0)}$.

If $v_1^\prime, \,v_2^\prime$ are two vertices of K ^′ with $ \lt v_1^\prime, \,v_2^\prime \gt \in (K^\prime){(1)}$ an edge joining them, and if at least one of the vertices is not in $\mathcal{S}$, then there is a lift of $ \lt v_1^\prime, \,v_2^\prime \gt $ to an invariant one-dimensional subcomplex of the saturation $G \lt v_1, \,v_2 \gt $ of any lift $ \lt v_1, \,v_2 \gt $ of $ \lt v_1^\prime, \,v_2^\prime \gt $. If $v_1^\prime, \,v_2^\prime \in \mathcal{S} $, then a lift of K ^′ is not a regular G-complex in general.

Lemma 4.4. Let K ^′ be a triangulation of $\Sigma_{\mathrm{g}^\prime}$ such that $\mathcal{S} \subset (K^\prime)^{(0)}$ and assume that the vertices $v_i, v_j \in \mathcal{S}$ do not span a simplex in K ^′. Then there exists a lift of K ^′ to a triangulation $K=\pi^{-1}(K^\prime)$ of $\Sigma_\mathrm{g}$ in which the action of G is simplicial and regular.

Proof. The existence of a lift is clear for the vertices. The lift of all them is a union of orbits. Consider a G-invariant Riemannian metric on $\Sigma_\mathrm{g}$. It induces a Riemaniann metric on $\Sigma_\mathrm{g}^\prime$ and thus it can be considered as a lift of the latter. If e ^′ is an edge joining the vertices $v_1^\prime, \, v_2^\prime$, then we can lift $v_1^\prime$ to $ v_1\in \Sigma_\mathrm{g}$ and take as a lift v ₂ of $v_2^\prime$ the closest point of the orbit $\pi^{-1}(v^\prime_2)$. Define a lift of $ \lt tv^\prime_1,(1-t) v^\prime_2 \gt $ as $ \lt tv_1,(1-t) v_2 \gt $. This gives a lift of e ^′ to an edge e joining v ₁ and v ₂. Its orbit $G \,e = G\, \lt v_1, v_2 \gt $ is G-homeomorphic to $ G \times_G \lt v_1, v_2 \gt $. The latter is a disjoint union of $|Gw$ edges if $ G_{v_1}= G_{v_2}= \textbf{e}$. If $G_{v_i} = \mathbb{Z}_m$ for one of $i=1, 2$, then it is a union of $\vert G\vert $ intervals grouped in $\vert G/G_{v_i} \vert $ disjoint sub-collections each consisting of $\vert G_{v_i} \vert $ edges intersecting at a single point of the orbit of v_i. This defines a triangulation $\pi^{-1}( \lt v^\prime_1, v^\prime_2 \gt )$ in which the action is simplicial and regular. Applying this procedure consecutively to all edges of K ^′, we get a graph $K^{(1)}$ on which the action of G is simplicial and regular.

Now we have to lift triangles of K ^′. If $\Delta^\prime = \lt v^\prime_1, v^\prime_2, v^\prime_3 \gt $ is a triangle of K ^′, then at most one of the vertices has a non-trivial isotropy group by our assumption. If all $ v^\prime_i \in \Sigma^\prime \setminus \mathcal{S}$ then π over Δ is a regular covering, and thus there exists a unique lift $\Delta= \lt v_1, v_1,v_3 \gt $ of $\Delta^\prime$ such that $\pi^{-1}(\Delta^\prime)$ G-homeomorphic to $G \times \Delta$ as a G-set. If for $\Delta^\prime = \lt v^\prime_1, v^\prime_2, v^\prime_3 \gt $ the isotropy group of one vertex is non-trivial, say $G_{v_1} = \mathbb{Z}_m$, then we have the lifts of the edges $ \lt v^\prime_1, v^\prime_2 \gt $ and $ \lt v^\prime_1, v^\prime_3 \gt $ to $ \lt v_1, v_2 \gt $ and $v_1,v_3$, respectively, by the procedure described above. The edge $ \lt v^\prime_2, v^\prime_3 \gt $ has a unique lift $ \lt v_2,v_3 \gt $ because π is a regular covering over $ \lt v^\prime_2, v^\prime_3 \gt $. This gives a lift of $\Delta^\prime$ to $\Delta = \lt v_1,v_2,v_3 \gt $. In this case, the saturation $G \Delta$ is G-homeomorphic to $G\times\Delta $, a union of $\vert G\vert $ triangles that are divided in $\vert G/G_{v_1} \vert $ disjoint sub-collections each consisting of $\vert G_{v_i} \vert $ triangles intersecting at a single point of the orbit of v ₁. Moreover, the group $G_{v_1}$ acts trivially on all triangles that meet at this point.

Consequently, we get a triangulation of $\Sigma_\mathrm{g}$ in which the action of G is simplicial and regular (even strictly regular), and the induced quotient triangulation $K/G$ is equal to K ^′.

Next we show the following lemma about a homogeneity of triangulations of a manifold with respect to the position of vertices.

Lemma 4.5. Let M be a smooth manifold of dimension $d \geq 2$ and K a triangulation of M with n vertices $v_1, \, v_2, \, \dots,\, v_n$. Then for every set $x_1, \, x_2, \, \dots\,,x_n \in M $ of pairwise pairwise different points in M there exists a triangulation K ^′ of M with vertices $\{x_1, \, x_2, \, \dots\,,x_n \}$ such that K ^′ is isomorphic to K.

Proof. The statement follows from the fact that a smooth manifold of $\dim \geq 2$ is n-tuples homogeneous for every $n\geq 1$ (cf. [23]). This means that for every two pairs of n-tuples $\{y_1, y_2, \, \dots, y_n\}$ and $\{x_1, x_2, \, \dots\,, x_n\}$ points of M there exists a diffeomorphism $\phi: M \to M$ such that $\phi(y_j)= x_j$. It is enough to take as $y_i\in M$ the images of $v_i \in K$ and the composition of the triangulating map with ϕ.

Before the formulation of the main theorem of this subsection, we need new notation. Let $\Sigma_{\mathrm{g}^\prime}=\Sigma_\mathrm{g}/G$ be the orbit surface of an orientation-preserving action of a group G with the branching data $(\mathrm{g}^\prime, r, m_1, \dots , m_r,)$. Let $\textbf{n} = \max\{n_{\mathrm{g}^\prime}, r\}$, and let $K^\prime(\textbf{n}) $ be a triangulation of $\Sigma_{\mathrm{g}^\prime}$ with n vertices given by theorem 4.3.

Theorem 4.6. Let $\Sigma_\mathrm{g}$ be an oriented surface of genus $\mathrm{g}$ with an orientation-preserving action by a finite group G. Denote by $\Sigma_{\mathrm{g}^\prime} = \Sigma_\mathrm{g}/G $ the quotient surface and let $r= \vert \mathcal{S}\vert $ be the number of singular fibers of the projection $\pi:\Sigma_\mathrm{g} \to \Sigma_{\mathrm{g}^\prime}$. Moreover, let be $n_{\mathrm{g}^\prime}$ the number defined in theorem 4.3, and let $\textbf{n} = \max(n_{\mathrm{g}^\prime}, r)$.

Then we have the following estimate for the number of orbits of minimal regular G-triangulation K of $\Sigma_\mathrm{g}$:

\begin{equation*} \textbf{n}\leq f_{G,0}(K) \leq \textbf{n} + \binom{r}{2} + \binom{r}{3} \,, \end{equation*}

with a convention that $\binom{r}{i}=0$ if $0\leq r \lt i$.

Proof of theorem 4.6

Let $K^\prime(\textbf{n}) $ be a triangulation of $\Sigma_{\mathrm{g}^\prime}$ with n vertices given by theorem 4.3. Using lemma 4.5, we can assume that r vertices of $K^\prime(\textbf{n})$ are r-points of $\Sigma_{\mathrm{g}^\prime}$ over which π is branched. We have to expand $K^\prime(\textbf{n}) $ to a triangulation $K^{\prime\prime}(\textbf{n}) \supset K^\prime(\textbf{n}) $ satisfying the assumption of lemma 4.4 by adding extra vertices, edges, and triangles if necessary.

Let $K^\prime_{min}(\textbf{n}) \subset K^\prime(\textbf{n})$ be the minimal subcomplex of $K^\prime(\textbf{n})$ containing all r-points of $\Sigma_{\mathrm{g}^\prime}$ over which π is branched, i.e. the subcomplex spanned by all such vertices.

We first assume that r > 3. If $r \lt n_{\mathrm{g}^\prime}$ then $K^\prime_{min}(\textbf{n}) $ is a proper subcomplex of $ K^\prime(\textbf{n})$, if $r\geq n_{\mathrm{g}^\prime}$ then $K^\prime_{min}(\textbf{n}) = K^\prime(\textbf{n})$. Consider the barycentric subdivision $ L:=(K^\prime_{min}(\textbf{n}))^\prime $ of $K^\prime_{min}(\textbf{n})$ and observe that all new vertices of L , i.e. the midpoints of edges and barycentres of triangles are in $ \Sigma_{\mathrm{g}^\prime} \setminus \mathcal{S}$. If $r\geq n_{\mathrm{g}^\prime}$ then we take $ K^{\prime\prime}(\textbf{n})= L$.

If $r \lt n_{\mathrm{g}^\prime}$ then we have to expand the simplicial complex structure of L to the required complex structure $ K^{\prime\prime}(\textbf{n}) $ of $\Sigma_{\mathrm{g}^\prime}$. Let $v_0 \in K^\prime(\textbf{n}) \setminus K^\prime_{min}(\textbf{n}) $ and $\Delta = \lt v_0, v_1, v_2 \gt $ be a triangle containing it. If $ \Delta \cap K^{\prime\prime}(\textbf{n}) = \emptyset $ or $\Delta \cap K^{\prime\prime}(\textbf{n}) = \{v_i\}$, $ i=1, 2 $ then we leave Δ not modified. On the other hand, if $ \Delta \cap K^{\prime\prime}(\textbf{n}) = \lt v_1, v_2 \gt $, then the barycentre $\tilde{v}=(\frac{1}{2}v_1, \frac{1}{2} v_2)$ is a vertex of L. In this case, we add to $K^\prime(\textbf{n})$ the edge $ \lt v_0, \tilde{v} \gt $ and consequently two new triangles for which it is the edge. In this way, we extend the simplicial structure of L to a new triangulation $K^{\prime\prime}(\textbf{n})$ of $\Sigma_{\mathrm{g}^\prime}$ satisfying the assumption of lemma 4.4. Observe that for this triangulation we add new vertices only for the barycentric subdivision L of $K^\prime_{min}(\textbf{n})$, but the number of vertices of L is equal to the number of vertices of $K^\prime_{min}(\textbf{n})$, number of its edges, and number of its faces, which is smaller or equal to r, $\binom{r}{2}$, and $\binom{r}{3}$, respectively. This gives the upper estimate of statement if $r\geq n_{\mathrm{g}^\prime}$. If $r \lt n_{\mathrm{g}^\prime}$ then we have to add remaining $n_{\mathrm{g}^\prime} - r$ vertices of $K^\prime(\textbf{n})$ which gives the upper estimate by $ (n_{\mathrm{g}^\prime} - r) + r + \binom{r}{2} + \binom{r}{3} = \textbf{n} + \binom{r}{2} + \binom{r}{3}$. It shows the estimate from above.

If r = 0 then the projection $\pi: \Sigma_\mathrm{g} \to \Sigma_{\mathrm{g}^\prime}$ is a regular cover, and the inequality becomes the equality. Similarly, if r = 1 then $K^\prime_{min}(\textbf{n})$ is a point and the condition of lemma 4.4 is satisfied for $K^\prime(\textbf{n})$, so that once more for the upper estimate holds.

If r = 2 then $K^\prime_{min}(\textbf{n})$ is either a union of two points and the condition of lemma 4.4 is satisfied, or an edge then we have to add only its centre to get L. Consequently, the inequality of statement holds also in this case. Finally, if r = 3 then $K^\prime_{min}(\textbf{n})$ is either the union of three points, the union of a point and an interval, or the connected union of two edges, or a triangle. Consequently to get L, we have to add to $K^\prime_{min}(\textbf{n})$ either one point or two points, or four points $= \binom{3}{1} + \binom{3}{2}$ points which once more confirms the statement.

The estimate from below is obvious, since a simplicial regular action on $\Sigma_\mathrm{g}$ induces a triangulation of $\Sigma_{\mathrm{g}^\prime}$.

Proposition 4.7. With the same notation and assumptions as in theorem 4.6 a similar (but weaker) estimates hold for $\mathrm{ct}_G(\Sigma_\mathrm{g})$, assuming that $g^\prime\ne 2$:

\begin{equation*} n_{\mathrm{g}^\prime} \leq \mathrm{ct}_G(\Sigma_\mathrm{g}) \leq \textbf{n} + \binom{r}{2} + \binom{r}{3} \, . \end{equation*}

Moreover, if $g^\prime=2$, then $\mathrm{ct}_G(\Sigma_g)\ge 9$.

Proof. The lower estimates follow from the inequality $ \mathrm{ct}_G(\Sigma_\mathrm{g})\geq \mathrm{ct}(\Sigma_{\mathrm{g}^\prime}) = n_{\mathrm{g}^\prime}$ if $\mathrm{g}^\prime \neq 2$. If $\mathrm{g}^\prime =2$, then by [2, proposition 3.4], we have $\mathrm{ct}(\Sigma^\prime_{\mathrm{g}^\prime}) = 9$.

The upper estimate follows from $\mathrm{ct}_G(X) \leq f_{G,0}(K)$ for any regular triangulation K of X.

We complete this section with a comparison of the values of the G-covering type and Lusternik–Schnirelman G-category for an orientation-preserving action of G on a surface $\Sigma_\mathrm{g}$. We have

Theorem 4.8 [15, theorem 4.11]

Let a finite group G act on $X=\Sigma_\mathrm{g}$ through orientation-preserving homeomorphisms. Then the set of all points is either the union of $r \geq 1 $ orbits or empty and we have: either $\mathrm{cat}_G(X) = cat(X/G) = 3$ and $\mathrm{g}^\prime \gt 0$ if the action is free, or

\begin{equation*}\mathrm{cat}_G(X) = \begin{cases}\; r\;\; {\text{if}}\;\; r \geq 3\,,\;\; {\text{or if}} \;\;r = 2\;\; {\text{and}} \;\; \mathrm{g}^\prime = 0\,,\cr \;3 \;\;{\text{if}}\;\; r \leq 2 \;\; {\text{and}}\;\; \mathrm{g}^\prime \geq 1\,,\;\; \end{cases}\;\;\end{equation*}

if the action is not free.

Remark 4.9. Note that if the action of G on $\Sigma_\mathrm{g}$ is free, then r = 0, and the estimate of theorem 4.6 reduces to the equality $f_{G,0}(\Sigma_\mathrm{g}) = n_{\mathrm{g}^\prime}$. Since $n_{\mathrm{g}^\prime}=\mathrm{ct}(\Sigma_{\mathrm{g}^\prime})$ (except for the case $\mathrm{g}^\prime = 2$, cf. [2]), we conclude $\mathrm{ct}_G(\Sigma_\mathrm{g})=\mathrm{ct}(\Sigma_\mathrm{g}/G)= n_{\mathrm{g}^\prime}$.

Moreover, it is known that there is a free action of the cyclic group $G=\mathbb{Z}_m$ on $\Sigma_\mathrm{g}$ if, and only if m divides $2-2\mathrm{g}$, and then $2-2\mathrm{g}^\prime = \frac{2-2\mathrm{g}}{m}$. This shows that $\mathrm{g}^\prime$ and $n_{\mathrm{g}^\prime}$ can be arbitrary large if $\mathrm{g}$ is large.

On the other hand, $\mathrm{cat}_G(\Sigma_\mathrm{g})= \mathrm{cat}(\Sigma_{\mathrm{g}^\prime}) = 3 $ for every such $\mathrm{g}$, as follows from theorem 4.8.

Observe that for a surface $\Sigma_\mathrm{g}$ with an action of G such that $ r \gt n_{\mathrm{g}^{\prime}}$, corollary 4.7 gives $\mathrm{ct}_G(\Sigma_\mathrm{g}) \geq r = \mathrm{cat}_G(\Sigma_{\mathrm{g}^\prime})$, i.e. the rate of growth is at least linear in r.

5. Estimate of G-covering type by G-genus

In this section, we give an estimate from below of the equivariant covering type expressed in terms of the equivariant genus γ_G analogous to the inequality of [13, theorem 2.2] in which the covering type is estimated by the Lusternik–Schirelmann category.

Theorem 5.1. Let X be a G-space which has a structure of a G-complex (or, more generally, a G-CW complex). Assume that the orbit types of the action of G on X are ordered linearly $(H_1) \geq (H_2) \geq \,\cdots\, \geq (H_k)$ (in particular, there is a unique minimal orbit type). Then

\begin{equation*} \mathrm{ct}_G (X) \,\geq\, \frac{1}{2} \, \gamma_G(X) \, (\gamma_G(X) +1)\,. \end{equation*}

Remark 5.2. Note that the assumption of theorem 5.1 is satisfied if the action is free or with one orbit type. It is also satisfied for every G-space X if G is a group whose subgroups are linearly ordered, e.g. if $G= \mathbb{Z}_{p^k}$ for p a prime. Moreover, if we assume that the regular good G-cover is strictly regular then in the hypothesis of lemma 5.3 the assumption about the order of orbit types is not necessary.

Let X be G-space with a structure of a G-complex and let $\mathcal{U}$ be a good G-cover of X. We can partition $\mathcal{U}$ into G-invariant subsets by taking orbits $\tilde U_i:= G U_i$. With this notation in mind, we have the following auxiliary result.

Lemma 5.3. If $\tilde{U}_1 \cap \,\cdots\, \cap \,\tilde{U}_{k+1} \neq \emptyset$ then $\mathrm{cat}_G({\overset{k+1}{\underset{i=1}\bigcup}}\, \tilde{U}_i) = 1$ and, consequently, $\gamma_G({\overset{k+1}{\underset{i=1}\bigcup}}\, \tilde{U}_i)=1$.

Proof. Using the G-nerve theorem, we can replace $Y= {\overset{k}{\underset{i=1}\bigcup}}\, \tilde{U}_i$ by a G-homotopy equivalent G-complex L. Moreover, by replacing, if necessary, an open set $U_i \subset \tilde{U}_i= GU_i$ by another element gU_i of the orbit, we can assume that ${U}_1 \cap \,\cdots\, \cap \,{U}_{k+1} \neq \emptyset$. Since $\{U_i\}_{i=1}^{k+1}$ form a good cover of Y, ${\overset{k+1}{\underset{i=1}\cup}}\, {U}_i$ is homotopy equivalent to a k-simplex $\sigma \subset L$ and $L=G \sigma= \tilde{\sigma}$ is its orbit. Note that for a regular simplicial action, the isotropy groups of points of faces of simplex are greater or equal to the isotropy groups of points of interior of simplex. Consequently, the minimal isotropy groups appear at the vertices of orbit of an simplex; therefore, $\tilde{\sigma}$ (the orbit of σ) can be described as the set

\begin{equation*} \left\{[t_1 \,\tilde{g}_1 v_1,\, \dots,\, t_{k+1} \, \tilde{g}_{k+1} v_{k+1}]\ \bigg| \; \; g\in G, \; \; \sum_{i=1}^{k+1}\, t_i=1\right\},\end{equation*}

where $\tilde{g}_i \in G/H_i$ is the class of $g_i\in G$ in $G/H_i$ and $H_i=G_{v_i}$ is the isotropy group of vertex v_i. Under this identification, σ corresponds to

\begin{equation*} \left\{[t_1 \,\tilde{e}_1 v_1,\, \dots,\, t_{k+1} \, \tilde{e}_{k+1} v_{n+1}]\ \bigg| \; \sum_{i=1}^{k+1}\, t_i=1 \right\} .\end{equation*}

Let $H=G_{v_{i}}$, for $1\leq i \leq k$, be the minimal orbit type on Y, i.e. on $ L=\tilde{\sigma}$. Without loss of generality, we may assume that i = 1. By the assumption about linear ordering of orbit types on X, thus on Y, for every $1 \leq i\leq k$, there exists a G-map $\phi_i: G/H_i \to G/H_1$ with $\phi_1=\mathrm{id}$. We can define a G-map $\phi: \tilde{\sigma} \to G/H_1 \times \sigma $ by

\begin{equation*}\phi([t_1 \,\tilde{g}_1 v_1,\, \dots,\, t_{k} \, \tilde{g}_{n+1} v_{n+1}])=[ t_1\, \alpha_1(\tilde{g}_1)v_1\,,\dots\,, t_{n+1}\, \alpha_{n+1}(\tilde{g}_{n+1})v_{n+1}].\end{equation*}

By taking the composition of ϕ with the projection onto $G/H_1$, i.e. $ p_1 \circ \phi: \tilde{\sigma} \to G/H_1$, and identifying $G/H_1$ with $Gv_1 \iota \tilde{\sigma}$, we get a G-equivariant map of $\psi = \iota \circ p_1\circ \phi:\tilde{\sigma} \to Gv_1$ onto the orbit. This map is a G-deformation retraction by the deformation $\psi = \iota \circ p_1\circ \phi_s$, where

\begin{align*}&\quad \phi_s([t_1\,\tilde{g}_1 v_1 \, ,t_2\,\tilde{g}_2 v_2\,, \dots,\,t_{k+1}\, \tilde{g}_{k+1} v_{k+1}])\\& = [t_1\,\tilde{g}_1 v_1 \, ,(1-s)t_2\,\tilde{g}_2 v_2\,, \dots,\,(1-s)t_{k+1}\, \tilde{g}_{k+1} v_{k+1}])\,,\end{align*}

which shows that $\mathrm{cat}_G(\tilde{\sigma}) = \mathrm{cat}_G({\underset{1}{\overset{n+1}{\cup}}}\, \tilde{V}_i)=1$.

Proof of theorem 5.1

Let X be a G-space whose strict G-covering type coincides with the G-covering type, let $\mathcal{U}= \{\tilde{U}_s\} $, $1\leq s\leq \mathrm{ct}_G(X)$ be its good G-cover and denote by K the nerve complex of $\mathcal{U}$. The vertices of K correspond to elements of $\mathcal{U}$ which are split into $\mathrm{ct}_G(X)$ orbits, and k-faces correspond to non-empty intersections $U_{i_1}\cap U_{i_2}\cap \, \cdots \, \cap U_{i_k}$. Note that $\gamma_G(|K|)=\gamma_G(X)$ and $\mathrm{ct}_G(|K|)= \mathrm{ct}_G(X)$.

We proceed by induction on $\gamma_G(X)$. If $\gamma_G(X) =1 $ then the right hand side of inequality is equal to 1, but $\mathrm{ct}_G(X)\geq 1 $, and the inequality is clearly satisfied.

Assume that the estimate holds for spaces with $\gamma_G(X) \leq n$, and let $\gamma_G(X) = n+1$. By our assumption on the order of orbit types and proposition 2.3, $\dim K= \dim (X/G)=\dim (X) \geq n$, so that there exist sets $U_1,\ldots, U_{n+1} \in \mathcal{U} $ which intersect non-trivially. Since $\mathcal{U}$ is regular, the orbits $\tilde{U}_s = G U_s$ are different for distinct i. Let $\tilde{U}:= \tilde{U}_1\cup\cdots \cup\, \tilde{U}_{n+1}$ and let $\tilde{U}^\prime$ be the union of orbits of the remaining elements of $\mathcal{U}$. By lemma 5.3, we have $\gamma_G(\tilde{V})=1$. Hence $\gamma_G(\tilde{V}^\prime)\geq \gamma_G(X)- \gamma_G(\tilde{V}) \geq n$ by the subadditivity of genus. If $\gamma_G(\tilde{V}^\prime) =n$ we can use the induction assumption to compute

\begin{equation*}\mathrm{ct}_G(X) \geq (n+1)+\frac{1}{2}\, n(n+1) = \frac{1}{2} \,(n+1)(n+2)\, .\end{equation*}

If $\gamma_G(\tilde{V}^\prime) =n+1$ we repeat the same argument subtracting each time n + 1 orbits $\tilde{U}_s$ from the good G-cover until the G-genus drops down by one and then we can use the induction assumption.

Remark 5.4. A similar strategy can be used to obtain an estimate of the G-covering type using the G-category. However, since the statement of proposition 2.3 does not hold in general for $\mathrm{cat}_G(X)$, the corresponding formula would be more complicated and thus less useful.

Example 5.5. Let $X:=S(V)$ be the sphere bundle of an $(n+1)$-dimensional complex free representation V of $G=\mathbb{Z}_p$. Then $\gamma_G(S(V))= \dim_{\mathbb{R}}(V)= 2n+2$ (cf. [1]) and $\mathrm{cat}_G(S(V))= \dim_{\mathbb{R}}(V)= 2n+2$ (cf. [22]). By theorem 5.1, we get

\begin{equation*}\mathrm{ct}_G(S(V)) \geq (n+1) \, (2n+3)\,.\end{equation*}

Since in this case $\mathrm{ct}_G(S(V)) = \mathrm{ct}(S(V)/G) = \mathrm{ct}(L^{2n+1}(p))$ we get the same as estimate of $\mathrm{ct}(L^{2n+1}(p))$ as the one given in [12] (and stronger than a previous estimate of [13]).

Observe that we may also reverse the above estimates to obtain upper bounds for the G-genus of a G-space based on the cardinality of some invariant regular good G-cover or on the number of vertices in an equivariant regular triangulation of a G-manifold.

5.1. Estimate of genus by the length index in the equivariant K-theory

In this subsection, we briefly recall the notion of an equivariant index based on the cohomology length in a given cohomology theory. It was introduced and described in details by Thomas Bartsch in [1, Chapter 4] and it can be used to estimate from below the G-genus of a G-space. We will use in for two generalized equivariant cohomology theories, the equivariant K-theory and the Borel cohomology.

First we consider the equivariant K-theory, denoted $K^*_G(X)$. It is generated by the K_G-theory, the equivariant K-theory of G-vector bundles, and is extended to a graded cohomology theory by the equivariant Bott periodicity (see Segal [24]).

Let us fix a set $\mathcal{A}$ of non-trivial orbits, which is obviously finite, since G is finite.

Definition 5.6. The $(\mathcal{A},K_G^*)$-cup length of a pair $(X, X^\prime)$ of G-spaces is the smallest r such that there exist $A_1, \,A_2, \,\ldots\, ,A_r \in \mathcal{A}$ and G-maps $\beta_i: A_i \to X$, $1\leq i \leq r$ with the property that for all $\gamma \in K^*_G(X,X^\prime)$ and for all $\omega_i \in \ker \beta_i^* $ we have

\begin{eqnarray*}\omega_1\, \cup\omega_2\cup\, \cdots \, \cup\omega_r \,\cup \, \gamma \,=\, 0\,\in K^*_G(X, X^\prime).\end{eqnarray*}

If there is no such r, then we say that the $(\mathcal{A},K_G^*)$-cup length of $(X,X^\prime)$ is $\infty$, and r = 0 means that $K^*_G(X, X^\prime)=0$. Moreover, the $(\mathcal{A}, K_G^*)$-cup length of X is by definition the cup length of the pair $(X,\emptyset)$.

Recall that for the equivariant cohomology theory $K^*_G$ and a G-pair $(X,X^\prime)$, the cohomology $K^*_G(X, X^\prime)$ is a module over the coefficient ring $K^*_G(\textrm{pt})$, via the natural G-map $p_X : X \to \textrm{pt}$. We write

\begin{eqnarray*}\omega \cdot \gamma= p_X^*(\omega) \cup \gamma \,\, \text{and} \,\, \omega_1 \cdot \omega_2 = \omega_1 \cup \omega_2,\end{eqnarray*}

for $ \gamma \in K^*_G(X,X^\prime)$ and $\omega_1, \, \omega_2 \in K^*_G(\textrm{pt})$. By taking $R:=K_{G}(\textrm{pt})=R(G)\subset K^*_G(\textrm{pt})$, we obtain the following version of [1, definition 4.1].

Definition 5.7. The $(\mathcal{A},K_G^*, R)$-length index of a pair $(X, X^\prime)$ of G-spaces is the smallest r such that there exist $A_1, \,A_2, \,\ldots\, ,A_r \in \mathcal{A}$ with the following property:

For all $\gamma\in K^*_G(X,X^\prime)$ and all $\omega_i \in R\cap\ker (K^*_G(\textrm{pt}) \to K^*_G(A_i))=\ker ( K_{G}(\textrm{pt}) \to K_G(A_i))$, $i=1, \, 2,\dots, r$, the product

\begin{eqnarray*}\omega_1 \cdot\omega_2 \cdots \omega_r\cdot\gamma =0\in K^*_G(X, X^\prime).\end{eqnarray*}

A comparison between the length index and the cup-length is given in the following statement (cf. Bartsch [1, p. 59]).

Proposition 5.8. For any system $\mathcal{A}$ and every pair of G-spaces $(X, X^\prime)$, we have

\begin{eqnarray*}(\mathcal{A},K_G^*, R)\,-\,{\text{length index of }}\; (X,X^\prime)\,\leq \,(\mathcal{A},K_G^*)\,-\,{\text{cup length of }} \; (X,X^\prime). \end{eqnarray*}

The $(\mathcal{A},K_G^*, R)$-length index has many properties which are important from the point of view of applications to study critical points of G-invariant functions and functionals (see [1]), and we are going to use some of them.

For the rest of this subsection let $G=\mathbb{Z}_{pn}$. Following [1], given two powers $m,n$ of p satisfying $ 1 \leq m\leq n\leq p^{k-1}$, we set

(5.1)\begin{equation}\mathcal{A}_{m,n}:= \{G/H\, | \, H \subset G;\, m\leq \vert H\vert \leq n\,\}, \end{equation}

where $\vert H\vert $ is the cardinality of H. Next we put

(5.2)\begin{equation} \ell_n(X, X^\prime) := (\mathcal{A}_{m,n},K_G^*,R)-{length\,index\,of} \;\; (X, X^\prime)\,. \end{equation}

Remark 5.9. By [1, observation 5.5], the index $\ell_n $ does not depend on m. It says that if $\mathcal{A}^\prime \subseteq \mathcal{A}$ is such that for each $A\in \mathcal{A}$ there exists $A^\prime \in \mathcal{A}^\prime$ and a G-map $A\to A^\prime$, then

\begin{eqnarray*} (\mathcal{A},K_G^{*},R)-{\text{length index}} \;=\; (\mathcal{A}^\prime,K_G^{*},R)-{\text{length index}}. \end{eqnarray*}

For us important is the following.

Proposition 5.10. (cf. [1, corollary 4.9]) Given a G-space X and a family of orbits $\mathcal{A}$, we have

\begin{equation*}\ell(X) \leq \gamma_G(X)\,,\end{equation*}

where the length $\ell$ in a generalized equivariant cohomology $h_G^*$ and the G-genus are taken relative to $\mathcal{A}$.

We will write $\mathcal{A}_X$ for the set of all G-orbits of X (up to isomorphism of finite G-sets).

Theorem 5.11 ([1, theorem 5.8])

Let V be an orthogonal representation of $G=\mathbb{Z}_{p^k}$ satisfying $V^G=\{0\}$, and let $d= \dim_{\mathbb{C}} V =\frac{1}{2}\, \dim_{\mathbb{R}} V$. Fix $m,\,n $ two powers of p as above. Then

\begin{eqnarray*} \ell_n (S(V))\, \geq \, \begin{cases} 1 + \Big[\frac{(d-1)m}{n}\Big]\;\;{\text{if}}\; \; \mathcal{A}_{S(V)} \subset \mathcal{A}_{m,n}, \cr \infty \phantom{+ \Big[\frac{(d-1)m}{n}\Big]} \;\;{\text{if}} \;\;\mathcal{A}_{S(V)} \nsubseteq \mathcal{A}_{1,n}\, , \end{cases} \end{eqnarray*}

where $[x]$ denotes the least integer greater than or equal to x. Moreover, if $ \mathcal{A}_{S(V)} \subset \mathcal{A}_{n,n}$, then

\begin{eqnarray*}\ell_n (S(V))= d\,.\end{eqnarray*}

As a direct consequence of theorems 5.1 and 5.11, we get the corresponding estimate of equivariant covering type and the number of vertices of a regular invariant triangulation of the orthogonal actions on spheres.

Theorem 5.12. Let V be an orthogonal representation of $G=\mathbb{Z}_{p^k}$, and $m,\, n, \, d $ as in theorem 5.11. If $\mathcal{A}_{S(V)} \subset \mathcal{A}_{m,n}$ then

\begin{equation*} \mathrm{ct}_G(S(V)) \geq \frac{1}{2} \, \bigl(1 + \Big[\frac{(d-1)m}{n}\Big]\bigr) \big(2 + \Big[\frac{(d-1)m}{n}\Big]\bigr).\end{equation*}

$\square$

Remark 5.13. Note that if $k\geq 2$ and m ≠ n, then S(V) from theorem 5.11 has more than one orbit type, in particular, it is not a free G-space.

Using theorem 5.12, we estimate the G-covering type in a slightly more complicated situation.

Proposition 5.14. Let $G=\mathbb{Z}_m$ be the cyclic group with $m= p_1^{k_1}\, p_2^{k_2}\,\cdots \, p_r^{k_r}$, p_i prime. Choose orthogonal representations $V_1,\ldots,V_r$ of G, such that, for each i, V_i is induced by a representation of $\mathbb{Z}_{p_i^{k_i}}$. Assume furthermore that $V_i^G=\{0\}$ for all i. Then

\begin{equation*}\mathrm{ct}_G(S(V_1\oplus V_2\, \oplus \cdots \, \oplus V_r))\,=\, \mathrm{ct}_{G_1}(S(V_1)) + \cdots + \mathrm{ct}_{G_r}(S(V_r))\,, \end{equation*}

where $G_i =\mathbb{Z}_{p^{k_i}}$ and $\mathrm{ct}_{G_i}(S(V_i))$ is estimated in theorem 5.12.

Proof. We show the statement in the case r = 2. The general case follows by the induction. Let $G= G_1\times G_2$ where $G_1=\mathbb{Z}_{p_1^{k_1}}$ and $ G_2 = \mathbb{Z}_{p_2^{k_2}}$. First observe that $S(V)^{G_1}= S(V_2)$, $S(V)^{G_2}= S(V_1)$, and $ S(V)= S(V_1) * S(V_2) $, where the action of $g=(g_1,g_2)$ on the element $[t v_1, (1-t)v_2]$ is defined as $g [t\, v_1, (1-t)\, v_2] = [t\, g_1 v_1, (1-t)\, g_2v_2]$.

Let $\mathcal{U}$ be a good G-cover of S(V) split into $\{\tilde{U}_j\}_{j=1}^n $, $ n= \mathrm{ct}_G(S(V))$ orbits. If $\tilde{U}_j \cap S(V)^{G_i} \neq \emptyset $, $i=1,2$, then $\tilde{U}_j$ is G-contractible to an orbit in $S(V_i)=S(V)^{G_{i}}$ as a G-map preserves fixed points of subgroups. By the same argument if $\tilde{U}_j \cap S(V)^{G_i} \neq \emptyset $, then $\tilde{U}_j \cap S(V)^{G_i} = \emptyset $ as $S(V)^{G_i} \cap S(V)^{G_i}= \emptyset $. Moreover, the families $\{\tilde{U}_j \cap S(V)^{G_i}\}_{j=1}^{n_i}$, $i=1,0$, $n_1+ n_2 \leq n$ form a good G-covers of $S(V)^{G_i}$ by definition 3.6 and proposition 3.8. This shows that $\mathrm{ct}_G(S(V)) \geq \mathrm{ct}_G(S(V_1)) +\mathrm{ct}_G(S(V_2))$, since $\mathrm{ct}_G(S(V_i)) = \mathrm{ct}_G(S(V)^{G_{1-i}})) = \mathrm{ct}_{G_i}(S(V_i))$, $i=1,2$ and proves the inequality in one direction.

To show the opposite inequality, we consider G-equivariant deformation retractions of the sets $W_1= \{[t\, v_1 +(1-t) v_2]:\, t\neq 1 \} $ onto $S(V_1) {\overset{G}\subset} S(V)$ and $W_2= \{[t\, v_1 +(1-t) v_2]:\, t\neq 0 \}$ onto $S(V_2) {\overset{G}\subset} S(V)$, denoted respectively as ρ ₁ and ρ ₂ and given by $\rho_1([t\, v_1 +(1-t) v_2])= v_1]$ and $\rho_2([t\, v_1 +(1-t) v_2])= v_2$.

Let $\mathcal{U}_i= \{\tilde{U}_s\}_{s=1}^{n_1}$, $i=1,2$, be a good G-cover of $S(V_i)$ with $n_i = \mathrm{ct}_{G}(S(V_i))$. Then the family of sets $ \mathcal{U}=\{\rho_1^{-1}(\mathcal{U}_1) \cup \rho_2^{-1}(\mathcal{U}_2)$ forms a good G-cover of S(V) of cardinality $n_1 +n_2$ as can be checked directly. This shows that $\mathrm{ct}_G(S(V)) \leq \mathrm{ct}_G(S(V_1)) +\mathrm{ct}_G(S(V_2))$ and consequently proves the statement.

Example 5.15. Let W be a d-dimensional orthogonal representation of the cyclic group $G=\mathbb{Z}_m$, such that the action of $G=\mathbb{Z}_m\subset S(\mathbb{C})$ as the roots of unity is free on S(W). Note that d can be arbitrary if m = 2, otherwise d must be even. Define $V:= W\oplus \mathbb{R}^1$; then $S(V)= S(W)*S(\mathbb{R})$ and the action of G on S(V) is has exactly two fixed points—the poles. Using the same construction as in the proof of theorem 5.12, one can show that

\begin{equation*}\mathrm{ct}_G(S(V) \leq \mathrm{ct}(S(\mathbb{R})) +\mathrm{ct}_G(S(W)) = 2 + \mathrm{ct}_G(S(W)).\end{equation*}

If $\dim W = 2 $, this leads to the inequality $\mathrm{ct}_G(S(V))\leq 2 + 3 = 5$.

On the other hand, if d = 2, then $\dim_{\mathbb{R}}(S(W)) =1$, and consequently $\dim S(V)=2$. Since this action on $S(V)=\mathbb{S}^2$ preserves the orientation, we can apply theorem 4.6 and obtain the equality $\mathrm{ct}_G(S(V))= \mathrm{ct}(S(V)/G) = \mathrm{ct}(\mathbb{S}^2)= 4$.

6. Cohomological estimates

There are several lower estimates of the equivariant Lusternik–Schnirelmann category $\mathrm{cat}_G(X)$ which are based on the multiplicative structure of the cohomology ring $\tilde{H}^*(X/G;R)$, or more generally of ${h_G}^*(X)$, where $h^*_G$ is any G-equivariant cohomology theory (see Bartsch [1], Marzantowicz [22]). This is due to the fact that cohomology products detect intersection patterns of subspaces of X. In this section, we will see that analogous ideas play an important role in the estimates of G-covering type and in the estimates of the number of orbits of regular triangulations of G-spaces. These estimates, when available, can considerably improve the category and genus estimates from the previous section. Additionally, this approach can be used even when the action of G has fixed points.

The singular cohomology theory has a natural filtration given as $H^{(s)}(X):= {\underset{k=0}{\overset{s} {\oplus}}}\, H^k(X)$ which is related to the geometrical filtration of X by its skeleta. In order to define an analogous filtration for a generalized equivariant cohomology theory $h^*_G$, we proceed as in Segal [24]. Let $h_G^*$ be a generalized equivariant cohomology theory. For a G-CW-complex X, we define a filtration of $h_G^*(X)$ as

\begin{equation*}h^*_{G,s} (X):= \ker ( h_G^*(X)\to h^*_G(X^{(s-1)})).\end{equation*}

This corresponds to the filtration of X by G-subspaces $X^s=\pi^{-1}(Y^{(s)})$, where $Y= X/G$ is the orbit space whose cellular structure is induced by the G-CW-structure of X. More generally, if X is a G-space for which there exists a G-simplicial complex K and a G-homotopy equivalence $w: X\to |K|$, then we take a filtration of X by G-subsets $X^{(s)}= w^{-1}(|K^{(s)}|$ and the corresponding filtration of $h^*_G(X)$ defined as above. The general case is discussed in Segal [24, Section 5] using the nerve of a G-stable closed finite covering of X, which is not necessarily a good cover. We are at position to define the degrees of element of $h_G^*(X)$.

The filtration of $h_G^*(X)$ defined above is decreasing ($d= \dim X$):

(6.1)\begin{equation} h_G^*(X)=h_{G,0}^*(X) \supset h^*_{G,1}(X) \supset\cdots \supset h^*_{G,d-1}(X) \supset h^*_{G,d}(X) = 0 \end{equation}

and $h_G^*(X)$ is a filtered ring, because

(6.2)\begin{equation} h_{G,s}^*(X) \, \cdot \, h_{G,s^\prime}^*(X)\,\subseteq h_{G,s+s^\prime}^*(X). \end{equation}

As a consequence, each $h_{G,s}^*(X)$ is an ideal in $h_G^*(X)$. In particular, we have the following characterization of $h_{G,1}^*(X)$ (cf. Segal [24, proposition 5.1(i), p. 146])

(6.3)\begin{equation} h_{G,1}^*(X) = \ker \left(h^*_G(X)\to \prod_{x\in X} h_G^*(G/G_x)\right) = \bigcap_{x\in X} \ker\big( h_G^*(X) \to h_G^*(G/G_x)\big). \end{equation}

Definition 6.1. The degree $|u|$ of an element $u\in h_G^*(X)$ is the maximal i, such that $u\in h_{G,i}^*(X)$.

Given an n-tuple of positive integers $i_1,\ldots,i_n \in {\mathbb{N}}$, we will say that a G-space X admits an essential $(i_1,\ldots,i_n)$-product in $h_G^*$ if there are cohomology classes $u_k \in h_{G,i_k}^*(X)$, such that the product $u_1\cdot u_2\cdots u_n$ is non-trivial. For every $(i_1,\ldots,i_n)$, there exist a G-space X and a generalized equivariant cohomology theory $h_G^*$ such that admits an essential $(i_1,\ldots,i_n)$-product. For example we can take $X=S^{i_1}\times\cdots\times S^{i_n}$, where $S^{i_n}= S(V_{i_n})$ is the sphere of an orthogonal orientation-preserving representation of G and $h_G^*(X):= H^*(X; \mathbb{Z})$. Clearly, if X admits an essential $(i_1,\ldots,i_n)$-product then so does every $Y{\overset{G}{\simeq}} X$, since their cohomology rings are isomorphic.

We may therefore define the G-covering type of the n-tuple of positive integers $(i_1,\ldots,i_n)$ with respect the equivariant cohomology theory $h_G^*$ as

\begin{equation*}\mathrm{ct}_G(i_1,\ldots,i_n):=\min\big\{\mathrm{ct}_G(X)\mid X \text{ admits an essential } (i_1,\ldots,i_n) \mathrm{-product}\big\}.\end{equation*}

The following proposition follows immediately from the definition.

Proposition 6.2.

\begin{equation*} \mathrm{ct}_G(X) \geq \max \{\mathrm{ct}_G(|u_1|,\ldots,|u_n|)\mid \ \mathrm{for\ all}\ 0\neq u_1\cdots u_n\in h^*_G(X)\}.\end{equation*}

Although the covering type of a specific product of cohomology classes may appear as a coarse estimate, it will serve very well our purposes. We will base our computations on the following technical lemmas. The first is a standard argument that we give here for the convenience of the reader. To shorten the notation, we denote $h_{G,1}^*(X)$ by $ \widetilde h^*_G(X)$ and note that our definition is equivalent to the following

\begin{equation*} \widetilde h^*_G(X) = {\underset{\phi: G {\overset{G}\to} X}{\bigcap}}\ker (\phi^*\colon h_G^*(X) \to h_G^0(G)).\end{equation*}

Lemma 6.3. Let $X=U\cup V$ where $U,V$ are open G-invariant subsets of X, and let $u,\,v\in\widetilde h^*_G(X)$ be cohomology classes whose product $u\cdot v$ is non-trivial. If U is G-categorical in X then $i_V^*(v)$ is a non-trivial element of $h_G^*(V)$ (where i_V denotes the inclusion map $i_V\colon V {\overset{G}{\hookrightarrow}} X$).

Proof. Assume by contradiction that $i_V^*(v)=0$. Exactness of the cohomology sequence

\begin{equation*}h_G^*(X,V) \stackrel{j_V^*}{\longrightarrow} h_G^*(X) \stackrel{i_V^*}{\longrightarrow} h_G^*(V)\end{equation*}

implies that there is a class $\bar v\in h_G^*(X,V)$ such that $j_V^*(\bar v)=v$. Moreover $i_U^*(u)=0$, because $i_U\colon U\hookrightarrow X$ is G-homotopic to the inclusion of an orbit $i_{Gx} \subset X$, so there is a class $\bar u\in h_G^*(X,U)$ such that $j_U^*(\bar u)=u$. Then $u\cdot v=j_U^*(\bar u)\cdot j_V^*(\bar v)$ is by naturality equal to the image of $\bar u\cdot\bar v\in h_G^*(X,U\cup V)=0$, therefore $u\cdot v=0$, which contradicts the assumptions of the lemma.

By inductive application of the above lemma, we obtain the following:

Lemma 6.4. Let $u_1,\ldots,\,u_n\in\widetilde h_G^*(X)$ be cohomology classes whose product $ u_1\cdots u_n$ is non-trivial, and let $X=U_1\cup\cdots\cup U_k\cup V$ where $U_1,\ldots,U_k$ are open, G-categorical subsets of X, and V is an open G-invariant subspace of X. Then the product of any $(n-k)$ different classes among $i^*_V(u_1),\ldots,i^*_V(u_n)$ is a non-trivial class in $h_G^*(V)$.

Towards the proof of the first statement, we begin the induction by using the following lemma.

Lemma 6.5. For $ u \in h^*_G(X)$, if $ |u| \geq i$ then $\mathrm{ct}_G(X) \geq i+2$.

Proof. Since $|u| \geq i$, we have $\dim(X) = \dim(X/G)\geq i$. From corollary 3.11, we get $\mathrm{ct}_G(i)\geq \mathrm{ct}(i) \ge i+2$. To complete the proof, we make an adaptation of [20, proposition 3.1]. Indeed, let K be a complex with simplicial regular action of G which is the nerve of a good regular G-cover $\mathcal{U}$ of X. By our assumption, $K/G$ is a complex of dimension $\geq i$. Let $\tilde{\mathcal{U}}^*$ be the induced good cover of the orbit space. If $\mathrm{ct}(|K|/G) = \mathrm{ct}(X/G)= i+1$, then the complex $K/G$ consists of one i-dimensional simplex $\sigma^*$. This means that $K =\pi^{-1}(\sigma^*) = G \sigma$ is the orbit of an i-dimensional simplex $\sigma =[v_0,\,v_1,\,\dots\,,v_{i}] \subset K$. We denote $ G \sigma$ by $\tilde{\sigma}$. Since for every subgroup $H\subset G$ the set K^H is a sub-complex of $K \tilde{\sigma}$ and $H^\prime \subset H$ implies $K^{H} \subset K^{H^\prime}$, the isotropy types of vertices are minimal (thus isotropy groups maximal!) and for any face of simplex $\tau \in K$ the isotropy types of each its face are smaller or equal to that of τ as the action is given by

\begin{equation*} g[t_0\,v_0, t_1\,v_1, \dots\,, t_{i}\,v_{i_1}]= [t_0\,gv_0, t_1\,gv_1, \dots\,, t_{i}\,gv_{i}]\end{equation*}

and G permutes vertices.

Let us form an invariant and closed cover of $K=\tilde{\sigma}$ by taking sets $\tilde{V}_j= G V_j$, where

\begin{equation*}V_j=\{[t_0\,v_0, t_1\,v_1, \dots\,, t_{i}\,v_{i}]: \sum_{j=0}^{i} t_j =1, \; \; {\text{and}}\; t_j \leq \frac{2}{3}\}\,.\end{equation*}

Observe that $\tilde{V}_j$ is G-contractible to $\tilde{v}_j = G v_j$. Moreover, for every k-tuple, $2\leq k \leq i$ the intersection $\cap \tilde{V}_j$ is G-contractible to the saturation of barycentre of the vertices corresponding to this tuple. In particular, $\tilde{V}_1\cap \tilde{V}_2$ is G-contractible to the saturation of $[\frac{1}{2} v_1,\frac{1}{2} v_2, 0, \dots,0]$.

It shows that for every non-trivial intersection $\tilde{h}_G^*(\bigcap \tilde{V}_j) = 0$. The Mayer–Vietoris sequence argument gives $\tilde{h}^*_G(K)=0$ which contradicts $|u| \geq i$. This shows that $\mathrm{ct}(|K|/G) \geq i+2$, and consequently $\mathrm{ct}_G(K)\geq i +2$.

We are ready to prove the main result of this section, an ‘arithmetic’ estimate for the covering type of an n-tuple:

Theorem 6.6.

\begin{equation*} \mathrm{ct}_G(i_1,\ldots\,i_n)\geq i_1 + 2\, i_2 + \, \cdots\, + n i_n + (n+1). \end{equation*}

Proof. We will induct on the number of elements in the sequence $(i_1,\ldots\,i_n)$. The case n = 1 is covered by lemma 6.5. Assume that the estimate holds for all sequences of length $(n-1)$ and consider classes $u_1\in\widetilde h_{G,i_1}^*(X),\ldots,u_n\in\widetilde h_{G,i_n}^*(X)$ such that the product $u_1 \cdots u_n \in h_{G,i_1+\ldots +i_n}^*(X)$ is non-trivial. By the equivariant nerve theorem 3.9 in every good G-cover $\mathcal{U}$, there exists a G-invariant subset $\mathcal{U}'{\overset{G}\subseteq} \mathcal{U}$, containing at least $(i_1+\cdots+i_n+1)$ orbits of sets that intersect non-trivially. Let us denote by U and V respectively the unions of elements of $\mathcal{U}'$ and of $\mathcal{U}-\mathcal{U}'$. Again by nerve theorem, the set U is G-contractible and thus G-categorical. Then the restriction of u_n on U is trivial, so by lemma 6.4 the restriction of $u_1\cdots u_{n-1}$ on V is non-trivial. By the inductive assumption $\mathcal{U}$ has at least

\begin{equation*}(i_1+2i_2+\ldots + (n-1)i_{n-1}+n)+(i_1+\ldots+i_n+1)=\end{equation*}

\begin{equation*}= i_1+2 i_2+\ldots+n i_n + (n+1), \end{equation*}

orbits of elements.

Remark 6.7. Note that the assumption of theorem 6.6 does not require any condition on the orbit types appearing in X. This extends essentially applications of this theorem.

As a first application of our results, let P(V) be the projectivization of a complex $(n+1)$-dimensional representation of G. The action of G on V induces an action on P(V), since $g(\lambda \, v) = \lambda \, g(v)$ for $\lambda \in \mathbb{S}^1 \subset \mathbb{C}$. The G-equivariant K-theory of P(V) can be described as

\begin{equation*}K_G^0(P(V)) = \mathbb{R}(G)[\eta]/e(V)\,,\end{equation*}

where $\mathbb{R}(G)$ is a representation ring of G and e(V) is an ideal in $\mathbb{R}(G)$ generated by the element $ {\underset{i=0}{\overset{n}\sum}} \, (-1)^i \wedge^i (V) \, \eta^{n+1-i}$ (see Bartsch [1] for details). Here η denotes a G-vector bundle conjugated to the G-Hopf bundle over P(V). Furthermore $K_G^1(P(V)) = 0$. By theorem 6.6, we have the following estimate.

Theorem 6.8. Let V be a complex representation of a finite group G of complex dimension $(n+1)$ and let P(V) be the projective space of V. Then

\begin{equation*} \mathrm{ct}_G(P(V)) \,\geq (n+1)^2\,.\end{equation*}

Proof. First note that from the description of $K_G^0(P(V))$ it follows that $ \eta^n= \eta \, \eta \, \cdots \,\eta \neq 0 $. Next we have to show that the degree $|\eta|$ is 2. A natural geometrical approach is to construct a G-CW-complex structure on P(V) in such a way that $P(V_1)\subset P(V_1\oplus V_2)\, \subset \cdots\, \subset P(V)$ is an ascending filtration of P(V) by G sub-complexes given by the decomposition $V= V_1\oplus \, \cdots\, V_s$ into irreducible factors. So for the 1-G-skeleton we have $P(V)^{(1)} \subset P(V_1)$. But then we have to do further analysis.

Słomińska [25] constructed an equivariant version of the Chern character

\begin{equation*}ch_G: K^j_G(X) \to {\underset{i=0}{\overset{\infty} {\, \oplus\,}}} H^{2i+j}_{Br,G}(X;\mathcal{K})\,,\, j = 0,\,1 \,,\end{equation*}

where $H^{2i+j}_{Br,G}(X;\mathcal{K})$ is the singular equivariant Bredon cohomology with the system $\mathcal{K}(G/H)= R(H)$ induced by the K_G-theory. By its construction, $ H^{2i+j}_{Br,G}(X;\mathcal{K})=0$ for $2i+j \gt \dim X$. Moreover, in [25], it is shown that

\begin{equation*} ch_G\otimes_{\mathbb{Q}} \textrm{id}_{\mathbb{Q}} : K^j_G(X)\otimes \mathbb{Q} \to {\underset{i=0}{\overset{\infty} {\, \oplus\,}}} H^{2i+j}_{Br,G}(X;\mathcal{K}\otimes \mathbb{Q} )\,\end{equation*}

is an isomorphism.

Let us show that $|e(V)| \geq 1 $. The restriction $e(V)|_{P(V)^{(0)}}$ to any orbit $Gx_0=G/G_{x_0}$, $Gx_0 \in P(V)^{(0)}$ is given by the restriction of η to the fiber at x ₀ (we view V as a representation of G_x). In other words, we may substitute η and the skew-symmetric powers of V to the formula defining e(V). This shows that $e(V)|_{P(V)^{(0)}}= 0$, i.e. $|e(V)|\geq 1$. Taking $X=P(V)^{(1)}$, we see that the image of $K^*_G(P(V))\to K_G^*(X)$ is the same as the image of $ {\underset{i=0}{\overset{\infty} {\, \oplus\,}}} H^{2i+j}_{Br,G}(P(V);\mathcal{K}\otimes \mathbb{Q} )$ in $ {\underset{i=0}{\overset{\infty} {\, \oplus\,}}} H^{2i+j}_{Br,G}(X;\mathcal{K}\otimes \mathbb{Q} )$. But $ {\underset{i=0}{\overset{\infty} {\, \oplus\,}}} H^{2i+j}_{Br,G}(X;\mathcal{K}\otimes \mathbb{Q} ) = H^{0}_{Br,G}(X;\mathcal{K}\otimes \mathbb{Q} )$, since $ \dim X =1$. On the other hand, $ch_G(\eta) = 0$ in $H^{0}_{Br,G}(X;\mathcal{K}\otimes \mathbb{Q} )$ as we showed above. Consequently, $e(V)|_{P(V)^{(1)}} = 0$ which means that $|e(V)|\geq 2$.

By theorem 6.6, we get $\mathrm{ct}_G(P(V)) \geq (n+1)^2 \,.$

The topological dimension $\dim P(V) $ is equal to $d= 2n$, so we can express the covering type of P(V) in terms of its geometric dimension as $\mathrm{ct}_G(P(V)) \geq \frac{(d+2)^2}{4}$.

Another class of examples to which our methods could be applied are $\mathbb{Z}_p^k$-actions on F_p-cohomology spheres and on spheres with different differential structures. In this situation, the most appropriate generalized cohomology theory is the Borel cohomology.

We begin with the following technical lemma.

Lemma 6.9. Let X be a compact G-CW-complex, such that $X^G \neq \emptyset$ and assume that X has a finite regular good G-cover $\mathcal{U} = \{\tilde{U}_i\}, \, i\in I$ of cardinality $|I|= \mathrm{ct}_G(X)$. Furthermore, let $\mathcal{U}^G$ be the subfamily of this cover consisting of $\{\tilde{U}_{i_s}\}, \, s\in S\subset I$ such that $\tilde{U}_{i_s}\cap X^G \neq \emptyset $.

Then the orbit of every element of this subfamily $\tilde{U}_{i_s} \in \mathcal{U}^G $ consists of one element $U_{i_s} = G U_{i_s}$, i.e. $U_{i_s}$ is G-invariant. Moreover, the cardinality $|S|= \mathrm{ct}_G(X^G) = \mathrm{ct}(X^G)$. Consequently, the cardinality of the set of orbits of family $\mathcal{U} \setminus \mathcal{U}^G$ is equal to $ |I|-|S|= |J|$, where $J=I\setminus S$.

Furthermore, $(X,X^G)$ is G-homotopy equivalent to a pair of regular G-complexes $(K, K^G)$ with the regular G-cover by the orbits of open stars of vertices $\mathcal{V}= \{\tilde{V}_i\}, \, i\in I$ and $\mathcal{V}^G= \{\tilde{V}_{i_s}\}, \, s\in S$, with $|I|=\mathrm{ct}_G(K)$,$|S|\geq \mathrm{ct}(K^G)$, for which

\begin{equation*} {\underset{j\in J}{\,\bigcup\,}} \tilde{V}_j \,=\, K\setminus K^G \,.\end{equation*}

Proof. The fact that each member of the subfamily $ {\underset{j\in I\setminus S}{\,\bigcup\,}} \tilde{V}_j \,=\, K\setminus K^G \,$ of $ \mathcal{U}^G $ is G-invariant follows from condition (RC1) of definition 3.5 of the regular cover.

The G-homotopy equivalence of the last part of statement follows from the G-nerve theorem (theorem 3.9), which also gives a one-to-one correspondence between $\mathcal{U} = \{\tilde{U}_i\}, \, i\in I$ and orbits of vertices of K, thus orbits of the good regular G-cover $\mathcal{V} = \{\tilde{V}_i\}, \, i\in I$ given by stars of vertices, and between $ \mathcal{U}^G$ and $\mathcal{V}^G$ respectively. From the construction of $\mathcal{V}$, it follows that $ {\underset{j\in I\setminus S}{\,\bigcup\,}} \tilde{V}_j \,=\, K\setminus K^G \,.$

To show that $|S|\geq \mathrm{ct}_G(X^G) = \mathrm{ct}(X^G)$, we use the G-cover of the already proved part of statement. Since the sets $ \{\tilde{U}_{i_s}\}_{s\in S} \cap |K^G|$ form a good cover of $|K^G|$, we have $|S| \geq \mathrm{ct}(|K|^G)$.

To study the equivariant covering type of G-spaces with a non-empty fixed point set, we need the relative, modulo X^G versions of $\mathrm{ct}_G$ and essential product. As before, we assume that X has a structure of a G-CW-complex.

Definition 6.10. For a pair $ X^G \subset X $, X a G-CW complex we define the relative strict G-covering type of the pair $(X, X^G)$ as

\begin{equation*} \mathrm{sct}_G(X, X^G) \,= \mathrm{sct}(X\setminus X^G) \,.\end{equation*}

First note that $\mathrm{sct}_G(X,X^G)$ is finite and $\mathrm{sct}_G(X,^G) +\mathrm{ct}(X^G) \leq \mathrm{ct}_G(X)$ as follows from lemma 6.9. We conjecture that

Conjecture 6.11.

\begin{equation*}\mathrm{sct}_G(X,X^G) +\mathrm{ct}(X^G) = \mathrm{ct}_G(X).\end{equation*}

Definition 6.12. Given an n-tuple of positive integers $i_1,\ldots,i_n \in {\mathbb{N}}$ we say that a G-space X with $X^G\neq \emptyset$ admits an essential relative $(i_1,\ldots,i_n)$-product in $h_G^*$ if there are cohomology classes $u_k \in h_{G,i_k}^*(X, X^G)$, such that the product $u_1\cdots u_n$ is non-trivial.

Clearly, if X admits an essential relative $(i_1,\ldots,i_n)$-product then so does every space Y that is G-homotopy equivalent to X, since their cohomology rings are isomorphic.

The following proposition follows immediately from the definition.

Proposition 6.13.

\begin{equation*} \mathrm{sct}_G(X, X^G) \geq \max \{\mathrm{ct}_G(|u_1|,\ldots,|u_n|)\mid 0\neq u_1\cdots u_n \in h^*_G(X,X^G)\}.\end{equation*}

From now on, we proceed as in the proof of first part of statement modifying each argument to the relative case. Note that $h_{G,1}^*(X,X^G) = \widetilde h^*_G(X,X^G) = h^*_G(X,X^G)$ as $\textrm{pt} {\overset{G}{\,\subset\,}} X^G {\overset{G}{\,\subset\,}} X$.

Lemma 6.14. Let $X\setminus X^G=U\cup V$ where $U,V$ are open and G-invariant in X, and let $u,\,u\in h^*_G(X,X^G)$ be cohomology classes whose product $u\cdot v$ is non-trivial in $h^*_G(X,X^G)$. If U is G-categorical in $X\setminus X^G$ then $i_V^*(u)$ is a non-trivial element of $h_G^*(V)$ (here i_V stands for the inclusion map $i_V: V {\overset{G}{\hookrightarrow}} X$).

Proof. Assume by contradiction that $i_V^*(u)=0$. Exactness of the cohomology sequence

\begin{equation*}h_G^*(X,X^G\cup V) \stackrel{j_V^*}{\longrightarrow} h_G^*(X,X^G) \stackrel{i_V^*}{\longrightarrow} h_G^*(V\cup X^G, X^G)\end{equation*}

implies that there is a class $\bar u\in h_G^*(X,V\cup X^G)$ such that $j_V^*(\bar u)=u$. Moreover $i_U^*(v)=0$, because $i_U\colon U\hookrightarrow X$ is G-homotopic to the inclusion of an orbit $i_{Gx} \subset X$, so there is a class $\bar v\in h_G^*(X,U\cup X^G)$ such that $j_U^*(\bar v)=v$. Then $u\cdot v=j_V^*(\bar u)\cdot j_U^*(\bar v)$ is by naturality equal to the image of $\bar u\cdot\bar v\in h_G^*(X,U\cup V)=0$, therefore $u\cdot v=0$, which contradicts the assumptions of the lemma.

By inductive application of the above lemma, we obtain the following:

Lemma 6.15. Let $u_1,\,\cdots,\, u_n\in h_G^*(X, X^G)$ be cohomology classes whose product $ u_1\cdots u_n$ is non-trivial, and let $X=U_1\cup\cdots\cup U_k\cup V$ where $U_1,\ldots,U_k$ are open, G-categorical subsets of X, and V is open invariant in X. Then the product of any $(n-k)$ different classes among $i^*_V(u_1),\ldots,i^*_V(u_n)$ is a non-trivial class in $h_G^*(V)$.

The above lemma implies the following relative version of theorem 6.6:

Theorem 6.16. Let $u_1,\,\ldots,\, u_n\in h_G^*(X, X^G)$ be cohomology classes whose product $ u_1\cdots u_n$ is non- trivial. Then

\begin{equation*} \mathrm{sct}_G(X,X^G)\geq i_1 + 2\, i_2 + \, \cdots\, + n i_n + (n+1). \end{equation*}

We are now ready to formulate a theorem which estimates $\mathrm{ct}_G$ or relative $\mathrm{sct}_G$ for an action of p-tori on a F_p-cohomology sphere.

Theorem 6.17. Let $\mathbb{S}^n$ be an n-dimensional manifold, homotopy equivalent to an F_p-cohomology sphere, with an action of the group $G=\mathbb{Z}_p^k$, with p-prime and $k\geq 1$. Assume first that $\mathbb{S}^G=\emptyset$. Depending on p, we have

\begin{equation*} \begin{matrix} \mathrm{ct}_G(\mathbb{S}^n) \geq \frac{(n+1)(n+2)}{2} \;\; {\text{if}}\;\; p=2\,,\cr{\phantom{\text{second row}}}\cr \mathrm{ct}_G(\mathbb{S}^n) \geq \frac{(d)(d+1)}{2}\;\; {\text{if}}\;\; p \gt 2, \;\;{\text{where}}\;\;d=\frac{n+1}{2}.\;\; \end{matrix} \end{equation*}

If $S^G \neq \emptyset$ then $S^G{\underset{F_p}{\,\sim\,}} \mathbb{S}^r $ is a F_p cohomology sphere of dimension $r\geq 0$ and

\begin{equation*} \begin{matrix} \mathrm{sct}_G(\mathbb{S}^n) \geq (r+2)+ \frac{(n-r-1)(n-r+2)}{2} \;\; {\text{if}}\;\; p=2\,, \cr{\phantom{\text{second row}}} \cr \mathrm{ct}_G(\mathbb{S}^n) \geq (r+2) + \frac{(d-1) (d+1)}{2}\;\; {\text{if}}\;\; p \gt 2, \;\;{\text{where}}\;\;d=\frac{n-r}{2} \,. \end{matrix} \end{equation*}

Proof. The proof is based on the Borel theorem (see [17, capt. IV] for an exposition). But for us a more convenient source of material is [9] where an approach to the Borel theorem given by tom Dieck in [26] is adapted to the problems discussed here.

The first part of the Borel theorem, which recovers the Smith theorem, states that for any action of the group $G=\mathbb{Z}_p^k$, or even more general for a finite p-group, we have that the fixed point set $ \mathbb{S}^G$ is F_p-cohomology equivalent to $\mathbb{S}^r$, where $-1\leq r \leq n$, i.e. it is an F_p-cohomology sphere of dimension $r\geq 0$, or it is empty if $r=-1$.

Let $\mathcal{H}+\mathcal{H}(\mathbb{S})$, denote the family $\{H_1, H_2, \, \dots\,, H_t\}$, $1\leq t\leq p^{k-1}$ of sub-tori $H \subset G$ of rank k − 1, such that $\mathbb{S}^{H_i}\neq \emptyset$, thus $\mathbb{S}^{H_i}\sim \mathbb{S}^{n_i}$, $n_i\geq 0$. The second part of Borel theorem states that

\begin{equation*} n-r\,= \sum_{i=1}^t \, (n_i -r) \,.\end{equation*}

Moreover, the theorem gives an information about the cohomology product in the Borel cohomology defined as $H^*_G(\mathbb{S};F_p) = H^*(EG{\underset{G}\times} X;F_p)$, where the latter is understood as the limit over filtration of the classifying space EG by its skeletons.

To show the first part of theorem, assume that $\mathbb{S}^G=\emptyset$, i.e. $r=-1$ Furthermore, the Borel theorem says that for an action with $\mathbb{S}^G = \emptyset$ we have $ H^*_G(\mathbb{S};F_p) = H^*(BG)/e(\mathbb{S})$ and $e(\mathbb{S})$ has the following description (cf. [9, theorem 3.1 and remark 3.2]). Namely, there exist elements $w_1,\,w_2, \, \dots\,, w_t \in H^*_G(\mathbb{S};F_2)$ such that $ e(\mathbb{S}) = w_1^{k_1} \cdot w_2^{k_2}\cdots w_t^{k_t} $ if p = 2, with $k_i = n_i-r$, or correspondingly elements $ s_1,\,s_2, \, \dots\,, s_t \in H^*_G(\mathbb{S};F_p)$ such that $ e(\mathbb{S})= s_1^{k_1} \cdot s_2^{k_2}\cdots s_t^{k_t} \neq 0$ if p > 2, with $k_i = \frac{n_i-r}{2}$, respectively. This means that the length of non-zero multiple of elements $w_1,\,w_2, \, \dots\,, w_t$, or $s_1,\,s_2, \, \dots\,, s_t$ correspondingly is equal to $n+1 -1 = n$, and $\frac{n+1}{2} -1 = \frac{n-1}{2}$ respectively, since $r=-1$.

In the case $r\geq 0$, i.e. if $(\mathbb{S}^n)^G\neq \emptyset$ we have similarly $ H^*_G(\mathbb{S}, \mathbb{S}^G;F_p) = H^*(BG)/e(\mathbb{S})$ with $ e(\mathbb{S}) = w_1^{k_1} \cdot w_2^{k_2}\cdots w_t^{k_t}\neq 0 $ if p = 2, but $ e(\mathbb{S})= s_1^{k_1} \cdot s_2^{k_2} \cdots s_t^{k_t} + \mathfrak{n} \neq 0$ if p > 2, where $\mathfrak{n}$ is nilpotent (cf. [9, theorem 3.1]). This means that the length of non-zero multiple of elements $w_1,\,w_2, \, \dots\,, w_t$, or $s_1,\,s_2, \, \dots\,, s_t$ correspondingly is equal to $n-r -1 $, or $\frac{n-r}{2} - 1 = \frac{n-r-2}{2}$ respectively.

In order to apply theorem 6.6, we have to show that the degree $|w_i| \geq 1$ and $|s_i| \geq 2$, respectively, but it follows from the description of elements $w_1,\,w_2, \, \dots\,, w_t$ and $s_1,\,s_2, \, \dots\,, s_t$, respectively. Remind that $H_G^*(*; F_p)= H^*(BG;F_p)$ and $H^*(B\mathbb{Z}_2^k; F_2)= F_2[\omega_1, \, \omega_2,\dots\,, \omega_k]$ is a polynomial ring, and $H^*(B\mathbb{Z}_p^k; F_p)= F_p[\gamma_1, \, \gamma_2\dots\,, \gamma_k] \otimes \wedge(\alpha_1\cdot\alpha_2\, \cdots\, \cdot \alpha_k)$ is the tensor product of polynomial ring R and the skew-symmetric algebra respectively. Furthermore, $w_i \in \ker (H^1(BG;F_p): {\overset{\iota^*}{\,\to\,}} H^1(BH_i; F_p)) $ induced by the map $\iota:BH_i \to BG$ and in the case of p-odd $s_i \in R\cup \ker (H^2(BG;F_p): {\overset{p^*}{\,\to\,}} H^2(BH_i; F_p)) $. In other words, $w_1,\,w_2, \, \dots\,, w_t$ and $s_1,\,s_2, \, \dots\,, s_t$ are images of elements described above by the homomorphism $p^*: H_G^*(X;F_p) \to H^*_G(pt) = H^*(BG;F_p)$ induced by the G-map $p: EG\times X \to EG$ on the orbit spaces. But in the singular cohomology theory $H^*(BG; F_p)$, the degree of an element is equal to its gradation, which shows that ω_i are of degree 1 and γ_i of degree 2. Consequently, their images and their linear combinations are of the same degree which shows that $|w_i| =1 $ and respectively $|s_i|=2$.

To complete the proof in the case where $\mathbb{S}^G=\emptyset$, it is enough to apply theorem 6.6 with the rank of summation from 1 to n and with the degree of each factor equal to 1 if p = 2, or with the rank of summation from 1 to $\frac{n-1}{2}$ and degree of each factor equal to 2 if p > 2.

In the case when $\mathbb{S}^G \neq \emptyset$, we have $\mathrm{ct}_G(X) \geq \mathrm{ct}(X^G) + \mathrm{sct}_G(X,X^G) = \mathrm{ct}(X^G) + \mathrm{sct}_G(X\setminus X^G)$ by lemma 6.9. Since $\mathbb{S}^G {\underset{F_p}{\,\sim \,}} \mathbb{S}^r$ is the F_p cohomology sphere of dimension r, we have $\mathrm{ct}(\mathbb{S}^G) = r+2$. To estimate the second term, we use theorem 6.16 and the above.