A geometric interpretation of Ranicki duality

Abstract

Consider a commutative ring $R$ and a simplicial map, $X\mathop {\longrightarrow }\limits ^{\pi }K,$ of finite simplicial complexes. The simplicial cochain complex of $X$ with $R$ coefficients, $\Delta ^*X,$ then has the structure of an $(R,K)$ chain complex, in the sense of Ranicki . Therefore it has a Ranicki-dual $(R,K)$ chain complex, $T \Delta ^*X$. This (contravariant) duality functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$ was defined algebraically on the category of $(R,K)$ chain complexes and $(R,K)$ chain maps.

Our main theorem, 8.1, provides a natural $(R,K)$ chain isomorphism:

\[ T\Delta^*X\cong C(X_K) \]

where $C(X_K)$ is the cellular chain complex of a CW complex $X_K$. The complex $X_K$ is a (nonsimplicial) subdivision of the complex $X$. The $(R,K)$ structure on $C(X_K)$ arises geometrically.

1. Introduction; description of results

This article is an addition to a theory of blocked surgery, pioneered by Ranicki, augmented by others in [1, 4, 5, 6, 8, 9, 16, 17], and still in a developing state.

Let $R$ be a commutative ring; let $K$ be a finite simplicial complex. In [16] Ranicki introduced the category of $(R,K)$ chain complexes and chain maps denoted $\mathcal {B} R_K$ here. He also defined algebraically, a contravariant functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$.¹

The simplest geometric example of an $(R,K)$ chain complex arises from a

$K$-space $(X,\pi )$. This is a finite simplicial complex $X$ and a simplicial map,

$\pi :X\to K$. In that case, the simplicial cochains on $X$ (with $R$ coefficients) form an $(R,K)$ chain complex denoted $\Delta ^*X$.

At the same time, $(X,\pi )$ specifies a regular CW complex $X_K$, which is a (non-simplicial) subdivision of $X$. We show that the cellular chain complex (with $R$ coefficients) of $X_K$ forms a second $(R,K)$ chain complex $C(X_K)$.

Our main theorem, theorem 8.1, exhibits a geometrically defined chain isomorphism between $C(X_K)$ and $T\Delta ^*X$. Roughly put:

\[ T\Delta^*X= C(X_K). \]

It is also our aim to give a transparent definition of this duality functor $T$, a clear treatment of Ranicki's natural transformation $e:T^2\to id$. and a simple proof that $e_C:T^2C\to C$ is an $(R,K)$ chain equivalence for all $C$.

Our larger goal is to facilitate applications of Ranicki's theory to geometric questions such as the topological rigidity of non-positively curved groups as in [4, 5, 9, 10] when those groups have elements of finite order.

The vehicle for such applications would be a full blown $K$-blocked surgery theory of which there are only hints in [16]. This would start with a degree-one normal map between closed manifolds, $(f, b): (M,\nu (M))\to (X,\xi )$ (as in [2]) together with a reference map, $\pi :X\to K$ as above. It would seek an $L$-theoretic obstruction to finding a normal cobordism of $(f,b)$ to a ‘K-blocked homotopy equivalence,’ $M'\to X$. But we will not pursue this here or even define the terms precisely.

In the classical case ($K=$ point; [2, 12, 18, 19, 20]) one has the ‘surgery obstruction’ $\sigma _*(f,b)\in L_n(\mathbb {Z}[\pi _1(X)])$ to such a normal cobordism. But this functor $L_n()$, was generalized in [16] to yield obstruction groups $L_n(\mathcal {A})$ for any ‘category-with-chain-involution’ $(A, *, \epsilon )$. Here $A$ is an additive category, $\mathcal {B} A\mathop {\to }\limits ^{*} \mathcal {B} A$ is a contravariant functor satisfying certain conditions, on the category $\mathcal {B} A$, of finite chain complexes in $A$, and $\epsilon :(*)^2\to id$, is an equivalence in the homotopy category of $\mathcal {B} A$.

Ranicki, in [16], then starts with a finite complex $K$ and a category with chain involution, $\mathcal {A}= (A, *, \varepsilon )$ as above. He then constructs the additive category $A_K$ of $K$-blocked objects from $A$, and $K$-blocked $A$-maps. From $*$, and $\varepsilon$, he defines the Ranicki Duality Functor $T:\mathcal {B}(A_K)\to \mathcal {B}(A_K)$, and the natural transformation $e: T^2\to id_{\mathcal {B}(A_K)}$. This construction allows one to define the surgery obstruction groups, $L_n(\mathcal {A}_K)$ where $\mathcal {A}_K =(A_K, T, e)$.

This seems to apply directly to a $K$-blocked normal map, $M^n\mathop {\longrightarrow }\limits ^{(f,b)} X^n\mathop {\to }\limits ^{\pi } K$. Here the relevant category seems to be $A=A(R)$, the category of finitely generated free modules over a fixed commutative ring $R$. We write $A R_K$ for $(A R)_K$ and $\mathcal {B} R_K$ for $\mathcal {B}(A R_K)$. Its objects are $(R,K)$-chain complexes. So the simplicial cochain complexes of $X$ and $M$ denoted $\Delta ^*X$ and $\Delta ^*M$, and the simplicial chain complexes, $\Delta X'$ and $\Delta M'$, are $(R,K)$-chain complexes. (See § 3). Thus the $L$-groups of $(\mathcal {A} R_K, T, e)$ seem likely to be useful.

However, Ranicki's definition of $\mathcal {B} R_K \mathop {\longrightarrow }\limits ^{T}\mathcal {B} R_K$ was only a starting point. Indeed his assertion in [16] of the crucial theorem that $(\mathcal {A} R_K, T, e)$ is a category with chain involution was only proved in 2018 (by Adams-Florou and Macko, [1]).

This paper interprets Ranicki's notions geometrically. Section 2 fixes chain-complex conventions. Section 3 reviews Ranicki's concepts concerning $(R,K)$ complexes while attempting to simplify notation. In § 5 we introduce the $(R,K)$ chain complex $C\otimes _KD$, defined if $D$ is an $(R,K)$ complex and $C$ is an $(R,K^{op})$ complex. This complex $C\otimes _KD$ is a certain quotient of $C\otimes _RD$.

Our definition (see 6.1) of the Ranicki dual $TC$, of an $(R,K)$ complex $C$, is:

\[ TC=C^*\otimes_K \Delta^*K. \]

In § 7 we show, using work of M. Cohen [3], that each $K$-space $(X,\pi )$ defines a certain regular CW-complex $X_K$, whose cellular chain complex has a natural $(R, K)$ structure. Therefore from each $K$-space $(X,\pi )$ we obtain three $(R,K)$ chain complexes:

1. (1) $\Delta ^*X$, the simplicial cochain complex of $X$ (definition 4.1).

2. (2) $C(X_K)$, the cellular chain complex of the CW complex $X_K$ (§ 8).

3. (3) $\Delta X'$, the simplicial chain complex of $X'$, the barycentric subdivision of $X$ (definition 7.2).

This paper shows that these three are closely related by $T$. Our main result, theorem 8.1, exhibits an isomorphism of $(R,K)$ chain complexes:

\[ \Phi_X: T\Delta^*X\cong C(X_K) \]

Then, using [13], we prove there are $(R,K)$ chain homotopy equivalences:

\[ T\Delta X'\simeq \Delta^*X;\quad C(X_K)\simeq \Delta X'. \]

When $X$ is a pl-manifold, and $C=C(X_K)$, Poincare duality then becomes an $n$-cycle in the $(R,K^{\mathrm {op}})$ complex, $Hom_{(R,K)}(TC, C)$.

This CW complex $X_K$ is a subdivision of $X$, and $X'$ is a simplicial subdivision of $X_K$. In fact, for each simplex $S$ of $X$ and each face $\sigma$ of $\pi (S)\in K$, there is a single cell $S_\sigma$ of $X_K$. Specifically, if $D(\sigma,\pi (S))$ is the dual cell of $\sigma$ in $\pi (S)$:

\[ S_\sigma=(\pi\mid S)^{{-}1}|D(\sigma,\pi(S))| \; . \]

The author is indebted to Jim Davis for his helpful comments. He also wants to acknowledge, with thanks, the referee's suggestions for improving the text.

2. Chain complex conventions

Throughout this paper, $R$ denotes a fixed commutative ring; $AR$ is the additive category of finitely generated free $R$ modules.

For any additive category $A$ we will write $\mathcal {B} A$ for the additive category of finite chain complexes, $C=\{C_q, \partial _q\}_{q\in \mathbb {Z}}$ and chain maps $f=\{f_q:C_q\to D_q\}_{q\in \mathbb {Z}}$ from $A$. (Finite means: $C_q=0$ for all but finitely many $q$). We abbreviate $\mathcal {B} (AR)$ to $\mathcal {B} R$.

As usual two chain maps $f,g: C\to D$ are chain homotopic if there is a sequence of $A$ maps, $h=\{h_q:C_q\to D_{q+1}\}$, for which $d^D_{q+1}h_q + h_{q-1}d^C_q = g_q-f_q \ \forall q$.

We regard $A$ as the full subcategory of $\mathcal {B} A$ consisting of chain complexes concentrated in degree zero.

Let $C,D\in Ob(\mathcal {B} R)$. The complexes $C\otimes _R D$, and $Hom_R(C,D)$ in $Ob(\mathcal {B} R)$, are:

\begin{align*} \boxed{(C\otimes_R D)_q = \sum_{r\in\mathbb{Z}} C_r\otimes_R D_{q-r};\quad Hom_R(C,D)_q =\sum_{r\in \mathbb{Z}}Hom_R(C_r, D_{q+r}) }\ \text{ and:}\\ \boxed{ d^{C\otimes D}(x\otimes y)= d^Cx\otimes y +({-}1)^{|x|} x\otimes d^Dy;\quad d^{Hom}\phi =d^D\circ \phi - ({-}1)^{|\phi| } \phi\circ d^C } \end{align*}

The evaluation map, $eval_{C,D}:Hom_R(C,D)\otimes _R C\longrightarrow D$ is the R-chain map:

\[ eval_{C,D}(f\otimes x)= f(x). \]

Note that $eval_{R,D}:Hom_R(R,D)\otimes _R R\cong D$.

Write $ev_C:C^*\otimes C\to R$ for $eval_{C,R}$.

The contravariant functor $\mathcal {B} R\mathop {\to }\limits ^{*} \mathcal {B} R$ is : $C^*= Hom(C,R); \; f^* = Hom(f, 1_R)$.

Therefore we have:

\[ (C^*)_{{-}q} = Hom_R(C_q,R); \quad d^{C^*}_{{-}q} = ({-}1)^{q+1}(d^C_{q+1})^*:(C^*)_{{-}q}\to (C^*)_{{-}q-1}. \]

The functor $*$ comes with a natural equivalence, $\varepsilon : (*)^2\to 1_{\mathcal {B} R}$. Specifically, the chain isomorphism $\epsilon _C: C^{**}\to C$ is characterized by the identity:

\[ a(\varepsilon_C(\alpha)) = ({-}1)^q \alpha(a) \quad \forall \alpha\in (C^{**})_{q},\; a\in (C^*)_{{-}q} . \]

3. Basic definitions for $(R,K)$ chain complexes

Definition 3.1 Let $K$ be a finite poset with partial order $\leq$.

$K^{op}$ denotes the same set with the opposite partial order.

(Later we will specialize to the case when $K$ is a finite simplicial complex).

1. (1) An (R,K) module is an ordered pair $M= (M(K), \{M(\sigma )\}_{\sigma \in K})$ such that:

   1. (a) $M(K)$ and each $M(\sigma )$ are $R$-modules in $Ob(A R)$;

   2. (b) $M(K) = \oplus _{\sigma \in K} M(\sigma )$.

   More generally, for any $S\subset K$ we write: $M(S)= \oplus _{\sigma \in S} M(\sigma ).$

2. (2) An $(R,K)$ map $M\mathop {\to }\limits ^{f}N$ of $(R,K)$ modules is a map $M(K)\mathop {\to }\limits ^{f} N(K)$ of $R$ modules, whose components, $f(\tau,\sigma ):M(\sigma )\to N(\tau ),$ satisfy:

   \[ f(\tau,\sigma)=\,0\text{ unless }\tau\geq\sigma. \]

3. (3) The additive category of $(R,K)$ maps and modules is written $AR_K$.

   We abbreviate the category of chain complexes, $\mathcal {B}(AR_K)$, to $\mathcal {B} R_K$.

4. (4) An object $C =\{ C_q, \partial _q \}_{q\in \mathbb {Z}}$ of $\mathcal {B} R_K$ is an ( $R,K)$ chain complex. We then write $C(K)$ for $\{C_q(K), \partial _q\}_{q\in \mathbb {Z}}$, an $R$-chain complex in $ob(\mathcal {B} R)$.

   Note: $C\in ob(\mathcal {B} R_K)$ is specified by specifying the $R$ complex $C(K)$ and the required collection $\{C_q(\sigma )\}_{\sigma \in K, q\in \mathbb {Z}}$ of $R$ submodules.

5. (5) Let $C,D\in ob(\mathcal {B} R_K).$ $Hom_{(R,K)}(C,D)$ is the $(R,K^{op})$ complex such that:

   1. (a) $Hom_{(R,K)}(C,D)(K)$ is the subcomplex of $Hom_R(C(K), D(K))$ given by those $f =\{f_q : C_q\to D_{q+|f|}\}_{q \in \mathbb {Z} }$ for which each $f_q$ is an $(R,K)$ map.

   2. (b) $Hom_{(R,K)}(C, D)_p(\sigma )$ is the set of $f\in Hom_{(R,K)}(C, D)(K)_p$ satisfying:

      \[ f_q \mid_{C_q(\tau) } =0\text{ if }\tau \neq \sigma,\; \forall\; q. \]

6. (6) We say a sequence of chain maps $0\to C'\mathop {\to }\limits ^{i} C\mathop {\to }\limits ^{j} C''\to 0$ in $\mathcal {B} R_K$ is exact if for each $\sigma \neq \tau$, $i(\sigma, \tau )=0,\; j(\sigma,\tau )=0,$ and, for all $q$, the corresponding sequence, $0\to C'_q(\sigma )\to C_q(\sigma )\to C''_q(\sigma )\to 0$. is an exact sequence in $\mathcal {A} R$. We then say $i$ is an $(R,K)$ monomorphism and $j$ is an $(R,K)$ epimorphism.

7. (7) Note that $*$ specifies a contravariant functor, $\mathcal {B} R_K\mathop {\longrightarrow }\limits ^{*}\mathcal {B} R_{K^{op}}$, provided that we define $(C^*)_q (\sigma )$ as $(C_{-q}(\sigma ))^*$ and $d^{C^*}$ as $d^{C(K)^*}$ for $C\in ob(\mathcal {B} R_K)$. $\mathcal {B} R_K\mathop {\longrightarrow }\limits ^{*}\mathcal {B} R_{K^{op}}$ preserves exactness and homotopy. The transformation $\varepsilon _C: C^{**}\to C$ of § 2 is an $(R,K)$ isomorphism, for all $C\in ob(\mathcal {B} R_K).$

8. (8) We say $S\subset K$ is full in K if, whenever $\rho, \tau \in S$, then:

   \[ \{\sigma\in K \mid \rho\leq \sigma\leq \tau\} \subset S. \]
   Let $C$ be an $(R,K)$ complex. Let $S$ be a full subset of $K$. We define $\partial _q^{C(S)}:C_q(S) \to C_{q-1}(S)$ by:
   \[ \partial_q^{C(S)} x = \sum_{\tau\in S}\partial^C(\tau, \sigma)x, \quad \forall x\in C_q(\sigma), \forall \tau, \sigma \in S. \]
   Then $C(S):=\{ C_q(S), \partial _q^{C(S)}\}_{q \in \mathbb {Z}}$ is an $R$ chain complex. But in many cases, it is neither a subcomplex nor a quotient complex of $C(K)$.

4. $K$ spaces and their chain complexes

For the rest of this paper, $K$ denotes a finite simplicial complex.

A simplicial complex $K$ is a poset so the above definitions apply. In this case $\sigma \leq \tau$ means that the simplex $\sigma$ is a face (not necessarily proper) of the simplex $\tau$.

$\Delta _*(K;R)=\{ \Delta _q(K;R),\partial _q\}_{q\in \mathbb {Z}}$ denotes the simplicial chain complex of $K$.

$\Delta ^*(K;R)=Hom_R(\Delta _*(X;R), R)$ denotes the simplicial cochain complex of $K$.

One can choose a basis, $bK$ for $\Delta _*(K;R)$ consisting of one oriented $q$-simplex, $\sigma =\langle v_0\dots, v_q\rangle \in \Delta _q(K;R)$ for each $q$-simplex with vertices $v_0,\dots v_q$, of $K$. Recall: $\langle v_0,\dots,v_q\rangle = sgn(\pi )\langle v_{\pi (0)},\dots,v_{\pi (q)}\rangle$ for each $\pi \in S_{q+1}$. The oriented $q$-simplex $\sigma \in \Delta _q(K;R)$ defines a dual cochain $\sigma ^*\in \Delta ^*(K;R)_{-q}$ such that $\sigma ^*(\tau ) =0$ for all $\tau \neq \pm \sigma$, and $\sigma ^*(\sigma )=1$.

One then defines $\sigma ^{**}\in \Delta _q(K;R)^{**}$ by: $\varepsilon (\sigma ^{**}) = \sigma$.

Each simplex $\sigma \in K$ defines subcomplexes, $\overline {\sigma }$ and $\partial \sigma$, and a subset $st(\sigma )$:

\[ \overline{\sigma}=\{\tau\in K\mid \tau\leq \sigma\}; \quad \partial\sigma=\{\tau\in K\mid \tau< \sigma\}; \quad st(\sigma)=\{\tau\in K\mid \tau\geq \sigma\} \]

The incidence number $[\tau, \sigma ]\in \{1,-1, 0\}$ is defined for any oriented simplices $\sigma, \tau$ of $K$. It satisfies: $\partial _q(\sigma )= \sum _{\tau \in bK} [\sigma, \tau ] \tau$ for any basis, $bK$ of oriented simplices of $K$. $[\sigma,\tau ]\neq 0$ iff $\tau$ is a codimension-one face of $\sigma$.

Definition 4.1 ($K$-spaces, $\Delta ^*X$ and $\Delta X$)

Let $K$ be a finite simplicial complex. A $K$-space is a pair $(X, \pi )$ where $X$ is a finite simplicial complex and $|X|\mathop {\to }\limits ^{\pi } |K|$ is a simplicial map, $X\to K$. A map of $K$-spaces, $(X,\pi _X)\to (Y, \pi _Y)$ is a simplicial map $f:|X|\to |Y|$ satisfying: $\pi _Y f=\pi _X$.

Let $(X,\pi )$ be a $K$-space.

$\Delta X$ denotes the $(R,K^{op})$ complex for which $\Delta X(K)= \Delta _*(X;R)$. For each $\sigma \in K$, $(\Delta X)_p(\sigma )$ is the submodule generated by oriented $p$-simplices in $\Delta _p(X;R)$ whose underlying p-simplex, $S\in X$, satisfies $\sigma = \pi (S)\in K$.

By definition, $\Delta ^*X= (\Delta X)^*$. Therefore $\Delta ^*X(K) = Hom_R(\Delta _*(X;R), R)=\Delta ^*(X; R)$, the simplicial cochain complex of $X$. For each $\sigma \in K$, $(\Delta ^*X)_{-p}(\sigma )$ is therefore the submodule spanned by all $S^*$ for which $S\in \Delta _p(X;R)$ is an oriented simplex and $\sigma =\pi (S)\in K$.

A map $f:X \to Y$ of $K$-spaces induces an $(R,K)$ chain map $f^*:\Delta ^*Y\to \Delta ^*X$ and an $(R,K^{op})$ chain map $f_*:\Delta Y\to \Delta X$.

The next lemma will be used in § 6.

Lemma 4.2 Suppose $S\in K$ and there is no  $\tau \in K$ for which $S<\tau$. The $K$-space $(\overline {S},inclusion)$ specifies the $(R,K)$ complex $\Delta ^*\overline {S}$. Then $\Delta ^*\overline {S}(st(\sigma ))$ is a contractible $R$-complex for all $\sigma \in K$ such that $\sigma \neq S$. Also $\Delta ^*\overline {S}(st(S)) = R S^*.$

Proof. It is obvious that $\Delta ^*\overline {S}(st(S)) = R S^*$ (after orienting $S$) and that $\Delta ^*\overline {S}(st(\sigma )) =0$ if $\sigma$ is not a face of $S$. So we assume $\sigma < S$. Let $\tau$ be the complementary face of $\sigma$ in $S$. Then the joins, $\overline {S} = \sigma * \tau$ and $\partial \sigma *\tau$ are contractible simplicial complexes. Note $st(\sigma )= \overline {S} -\partial \sigma *\tau$. Consequently, $\Delta ^*\overline {S}(st(\sigma )) = \Delta ^*(\sigma * \tau, \partial \sigma * \tau ;R)$ is a contractible chain complex.

5. $C\otimes _K D$ and the isomorphism $Hom_{(R,K)}(D, C^*) \cong (C\otimes _K D)^*$

Throughout this section, $C$ denotes an $(R,K^{op})$ complex and $D$ denotes an $(R,K)$ complex.

We will first define two $(R,K)$ complexes: $C\otimes _R D$ and a quotient of this, ${C\otimes _K D}$.

In $K$, the star of any simplex , $st(\sigma )$, as well as $K-st(\sigma )$ are full in $K$. Moreover the chain complex $C(K-st(\sigma ))$ is a subcomplex of $C(K)$ and $C(st(\sigma ))$ is a quotient complex. These fit into a short exact sequence of chain maps in $\mathcal {B} R$:

\[ 0\to C(K-st(\sigma))\mathop{\longrightarrow}\limits^{i_\sigma} C(K)\mathop{\longrightarrow}\limits^{p_{st(\sigma)}}C(st(\sigma)) \to 0 \]

Here $C(K)\mathop {\longrightarrow }\limits ^{p_{st(\sigma )}}C(st(\sigma ))$ is defined by: $p_{st(\sigma )}|_{C_q(st(\sigma ))}= 1_ {C_q(st(\sigma ))}$; and $p_{st(\sigma )}|_{C(K-st(\sigma ))}=0$.

(For the $(R,K)$ complex $D$, we get $0\to D(st(\sigma ))\to D(K)\to D(K- st(\sigma )){\to 0}$).

Definition 5.1 ($C\otimes _KD$, $C\otimes _RD$, and $C\otimes _RD \mathop {\longrightarrow }\limits ^{\pi _{C,D}}C\otimes _K D$).

Let $C$ be an $(R,K^{op})$ complex and $D$ be an $(R,K)$ complex.

1. (1) Let $C\otimes _RD$ be the $(R,K)$ complex for which:

   \begin{align*} & (C\otimes_RD)(K)= C(K)\otimes_R D(K); \\ & (C\otimes_R D)_q(\rho)= (C(K)\otimes_R D(\rho))_q \quad \forall \rho\in K, q\in \mathbb{Z} \end{align*}

2. (2) Let $C\otimes _KD$ be the $(R,K)$ complex for which:

   1. (a) $(C\otimes _K D)_q(K)= \sum _{\rho \in K} (C(st(\rho ))\otimes _R D(\rho ))_q \ \forall q\in \mathbb {Z}$

   2. (b) $(C\otimes _K D)(\rho )= C(st(\rho ))\otimes _R D(\rho ) \ \forall \;\rho \in K$

   3. (c) The map $C\otimes _RD\mathop {\longrightarrow }\limits ^{\pi _{C,D}} C\otimes _KD$ is an $(R,K)$ chain epimorphism, if we define $\pi _{C,D}$ by requiring that $\pi _{C,D}(\sigma,\rho )=0$ for $\sigma \neq \rho$ and:

      \[ \pi_{C,D}(\rho, \rho)= p_{st(\rho)}\otimes_R 1_{D(\rho) }:C(K)\otimes_R D(\rho) \longrightarrow C(st(\tau))\otimes _R D(\rho). \]

Expicitly, for any $\rho \leq \tau$ and $x\otimes _Ry\in C_r(\tau )\otimes _RD_{q-r}(\rho )\subset (C\otimes _KD)_q(\rho )$, we have

(5.1)\begin{equation} \boxed{ d^{C\otimes_K D}(x\otimes y) = \sum_{\{\sigma| \rho\leq \sigma\leq\tau\} } d^C(\sigma,\tau)x \otimes y + ({-}1)^r x\otimes d^D(\sigma, \rho) y. } \end{equation}

We now show that $(C\otimes _KD)^*$ is a convenient expression for $Hom_{(R,K)}(D, C^*)$:

Lemma 5.2 There is a natural isomorphism $\Psi$ of functors, denoted,

(5.2)\begin{equation} \Psi_{C, D}: Hom_{(R,K)} (D, C^*)\cong (C\otimes_K D)^* \end{equation}

for any $(C,D)\in Ob(\mathcal {B} R_{K^{op}}\times \mathcal {B} R_{K})$.

Proof. Suppose $f$ is in $Hom_{(R,K)}(D, C^*)_q(\sigma )$ for some $\sigma \in K$ and $q\in \mathbb {Z}$. Define an $R$-map, $\Psi (f): C(st(\sigma ))\otimes D(\sigma )_{-q}\to R$, by the formula:

\[ \Psi(f)(x\otimes y)= ({-}1)^{|x| |y|} f(y)(x) \quad \text{for }x\otimes y\in (C\otimes_KD)_{{-}q}(\sigma). \]

The same formula yields $0$, if $x\otimes y$ is in $(C\otimes _KD)(\tau )_{-q}$ for $\tau \neq \sigma$. One easily sees that this rule (i.e. $f\mapsto \Psi (f)$ gives an isomorphism,

\[ \Psi_{C,D}: Hom_{(R, K)}(D,C^*)\mathop{\longrightarrow}\limits^{{\cong}} (C\otimes_K D)^* \]

of $(R,K^{op})$ complexes for all $(C,D) \in Ob(\mathcal {B} R_{K^{op}}\times \mathcal {B} R_{K})$. Naturality is obvious.

6. Ranicki Duality and the $(R,K)$ chain equivalence $e:T^2\to 1_{\mathcal {B} R_K}$

Definition 6.1 Ranicki Duality is the contravariant functor $\mathcal {B} R_K\mathop {\longrightarrow }\limits ^{T}\mathcal {B} R_K$ defined for a chain complex $C\in Ob(\mathcal {B} R_K)$ and a $(R,K)$ chain map, $f:C\to D$ by:

\[ TC= C^*\otimes_K\Delta^*K\quad Tf= f^*\otimes_K 1_{\Delta^*K} \]

$\Delta ^*K$ comes from the $K$-space, $(K,1_K)$. After examining [16], p. 75 and p. 26, lines -6 to -4 one can see that this is in agreement with the definition indicated there, up to isomorphism and differences in sign conventions. In particular compare our formula for $d^{C\otimes _KD}$ with that on p.26, line -5 of [16].

Corollary 6.2 $T$ is an exact homotopy functor.

Proof. By lemma 5.2, $TC= C^*\otimes _K \Delta ^* K$ is isomorphic to $Hom_{(R,K)}(\Delta ^*K, C)^*$ (since $\varepsilon _C:C^{**}\cong C$ for all $C$). But $C\mapsto C^*$ and $C\mapsto Hom(\Delta ^*K, C)$ are both exact homotopy functors. The result follows.

We now want to show that $T^2C$ and $C$ are $(R,K)$-chain equivalent. See 6.5.

Definition 6.3 (of $E_C:Hom_{(R,K)}(\Delta ^*K, C)\otimes _K \Delta ^*K\to C$).

Let $C$ be an $(R,K)$ complex.

Consider the evaluation chain map, $eval_{A,B}:Hom_R(A,B)\otimes _R A\to B$, when $A=\Delta ^*K(K)$ and $B=C(K)$. Its restriction to $(Hom_{(R,K)}(\Delta ^*K, C)\otimes _R\Delta ^*K)(K),$ denoted $E'_C,$ is an $(R,K)$ chain map,

\[ E'_C:Hom_{(R,K)}(\Delta^*K, C)\otimes_R\Delta^*K\to C \]

(by definition of an $(R,K)$ map). Moreover, for each $\sigma \in K$, $E'_C$ annihilates $Hom_{(R,K)}(\Delta ^*K, C)(K-st(\sigma ))\otimes _R\Delta ^*K(\sigma )$. Therefore $E'_C$ descends uniquely to an $(R,K)$ chain map,

\[ E_C:Hom_{(R,K)}(\Delta^*K, C)\otimes_K\Delta^*K\to C,\quad E_C(f\otimes \sigma^*)=f(\sigma^*). \]

satisfying: $E'_C= E_C\circ \pi _{H, \,\Delta ^*K}$ . Here $H= Hom_{(R,K)}(\Delta ^*K, C)$ (see 5.1).

$E$ is obviously natural in $C$.

For each $(R,K)$ complex $C$, define

\[ \Psi_{C^*}= \Psi_{C^*, \Delta^*K}:Hom_{(R, K)}(\Delta^*K,C^{**})\mathop{\longrightarrow}\limits^{{\cong}} (C^*\otimes_K \Delta^*K)^* \]

In view of lemma 5.2. we have an $(R, K)$ chain isomorphism:

\[ \Psi_{C^*}\otimes 1_{\Delta^*K}: Hom_{(R,K)}(\Delta^*K, C^{**})\otimes_K\Delta^*K\mathop{\longrightarrow}\limits^{{\cong}}(C^*\otimes_K\Delta^*K)^*\otimes_K \Delta^*K= T^2C . \]

Definition 6.4 For each $(R,K)$ complex $C$ define $e_C:T^2C\to C$ by

\begin{align*} e_C& = \varepsilon_C\circ E_{C^{**}}\circ (\Psi_{C^{*}}\otimes 1_{\Delta^*K})^{{-}1}:\\ & (C^{*} \otimes_K \Delta^*K)^*\otimes_K \Delta^*K\to Hom_{(R,K)}(\Delta^*K, C^{**})\otimes_K\Delta^*K\to C^{**}\to C. \end{align*}

Note $e_C$ is an $(R,K)$ chain epimorphism and $e$ is a natural transformation.

Theorem 6.5 $e_C: T^2C\longrightarrow C$ is an $(R,K)$ chain equivalence, for each $(R,K)$ complex $C$.

Proof. By [16] (proposition 4.7), we need only prove that $e_C(\sigma,\sigma ): T^2C(\sigma ) \to C(\sigma )$ is an $R$-chain equivalence, for all $\sigma \in K$. (No proof of this proposition appears in [16]. A brief proof appears in Appendix 2).

Case I: Assume there is a simplex $S\in K$ for which: $C(\sigma )=0 \ \forall \;\sigma \neq S$.

We need only show $e_C(S,S)$ is a chain isomorphism, and $T^2C(\sigma )$ is contractible for $\sigma \neq S$. We compute, for all $\sigma \in K$, in view of the restriction on $C$:

\begin{align*} TC(st(\sigma)) & = (C^*\otimes_K \Delta^*K)(st(\sigma)) = (C^*\otimes_R \Delta^*\overline{S})(st(\sigma)) \\ & =C^*(S)\otimes_R\Delta^*\overline{S} (st(\sigma))\\ \text{So: } T^2C(\sigma) \cong & C^{**}(S)\otimes_R \Delta^{**}\overline{S} (st(\sigma)) \otimes_R R\sigma^* \end{align*}

So for $\sigma \neq S, \; T^2C(\sigma )$ is contractible because $\Delta ^{**}\overline {S} (st(\sigma ))$ is contractible by 4.2.

Next we prove that the map

\[ e_C(S,S)= \varepsilon_C(S,S)\circ E_{C^{**}}(S,S)\circ(\Psi_C^*\otimes 1_{\Delta^*K})^{{-}1} (S,S) \]

is an isomorphism, or equivalently that $E_C(S,S)$ is an isomorphism.

Assume $S$ has been oriented. Because $C(\sigma )=0$ for $\sigma \neq S$,

\[ E_C(S,S): [Hom_{(R,K)}(\Delta^*K, C)\otimes_K\Delta^*K](S)\to C(S) \]

is simply: $eval_{RS^*, C(S)}:Hom_R(RS^*, C(S))\otimes _R RS^*\to C(S)$.

This is a chain isomorphism as observed in § 2. So $e_C(\sigma,\sigma )$ is a chain isomorphism for $\sigma = S$ and a chain equivalence for $\sigma \neq S$. This completes the proof in Case I.

Case II (the general case): For any $C\neq 0$ in $\mathcal {B} R_K$ one can choose some $S\in K$ for which $C(S)\neq 0$, and an exact sequence $0\to C'\mathop {\longrightarrow }\limits ^{i} C\mathop {\longrightarrow }\limits ^{j} C''\to 0$ for which $i(S,S):C'(S)\to C(S)$ is an isomorphism, and $C'(\sigma )= 0$ for $\sigma \neq S$. For example, choose $S$ to be of maximum dimension among $\{\sigma \in K \mid \; C(\sigma )\neq 0\}$).

The argument is by induction on the number $n$, of $\sigma \in K$, for which $C(\sigma )\neq 0$.

If $n=1$, Case I applies. If $n>1$, by induction, $e_{C''}(\sigma,\sigma )$ and $e_{C'}(\sigma,\sigma )$ are $R$ chain equivalences. Also the commuting diagram below has exact rows.

Therefore $e_C(\sigma,\sigma )$ is an $R$-chain equivalence for all $\sigma$. This completes the proof.

Note: The first proof of the above theorem appeared in [1].

7. Construction of the ball complex $X_K$

The purpose of this section is to construct the complex $X_K$ advertised in the introduction and establish its properties.

Definition 7.1 (of $X'$): Let $X$ be a finite simplicial complex in a euclidean space, with vertex set $V_X$. Its underlying polyhedron is: $|X|=\cup \{\sigma \;| \;\sigma \in X\}$. For each $p\geq 0$, $X_p$ denotes the set of $p$-simplices of $X$.

If $|X|$ is pl-homeomorphic to $I^n$ we say $|X|$ or $X$ is a pl $n$-ball and write $\partial X$ for the subcomplex for which $|\partial X|= \partial |X|$.

Each p-simplex $\sigma \in X$ is the convex hull, $[v_0, v_1,\dots, v_p]$, of its vertices in $V_X$. Its barycenter is $\hat {\sigma }:= \frac {1}{p+1} \sum _{i=0}^p v_i \in \sigma ^\circ$.

Choose a point $b\sigma \in \sigma ^\circ$, the interior of $\sigma$, for each $\sigma \in X$.

The derived complex $X'$ is defined as the unique simplicial subdivision of $X$ for which $V_{X'} = \{b\sigma \mid \; \sigma \in X\}$. $X'$ has one $p$-simplex, $[b\sigma _0, b\sigma _{1} \dots b\sigma _p]$, for each decreasing sequence of simplices $\sigma _0>\dots >\sigma _p$ of $X$.

If $\sigma _0>\dots >\sigma _p$, the ordered $p+1$ tuple $(b\sigma _0, b\sigma _{1}, \dots, b\sigma _p)$ then specifies an oriented $p$-simplex in $\Delta _p(X';R)$ which we denote $\langle \sigma _0, \sigma _{1} \dots, \sigma _p\rangle$ (suppressing the barycenters for concision).

These form a canonical basis for $\Delta _p(X';R)$ (in contrast to $\Delta _p(X;R)$).

Because we want to use the McCrory cap product, we follow the orderings of [13] regarding simplices of $X'$.

Definition 7.2 (of $\Delta X'$): Let $(X,\pi )$ be a $K$-space. The derived complexes of $(X,\pi )$ are the simplicial subdivisions $X'$ of $X$, and $K'$ of $K$ whose vertex sets $\{b\sigma \mid \sigma \in K\}$ and $\{bS\mid S\in X\}$ are chosen as follows:

\[ \text{If } \sigma\in K, \quad b\sigma:= \hat{\sigma}\in\sigma^\circ; \]

\[ \text{If } S\in X\text{ and } \sigma=\pi(S), \quad bS:=\text{ centroid of }(S\cap \pi^{{-}1} (\hat{\sigma}))\in S^\circ. \]

By construction, $\pi (V_{X'})\subset V_{K'}$. So $\pi$ is also a simplicial map from $X'$ to $K'$, because $\pi$ is linear on each simplex of $X'$.

$X'$ provides a second geometric example, $\Delta X'$, of an $(R,K)$ complex:

We define $\Delta X'$ by,

1. (1) $\Delta X'(K)= \Delta _*(X';R)$.

2. (2) For each $\sigma \in K, p\in \mathbb {Z},\;\; (\Delta X')_p(\sigma )$ is the submodule of $\Delta _p(X';R)$ spanned by all $\langle Q^0,\dots Q^p\rangle$ in $X'$ for which $\sigma = \pi (Q^p)$.

It is straightforward to see that $\Delta X'$ is an $(R,K)$ complex.

The dual cone of a simplex $\sigma \in K$, denoted $D(\sigma, K)$, is a subcomplex of $K'$ first defined in [15], § 7. It is a pl ball if $K$ is a pl-manifold). It gives rise to several ‘dual’ subcomplexes in $K'$ and $X'$ which we define now.

Definition 7.3 Let $(X,\pi )$ be a $K$-space. Suppose $\sigma, \tau \in K$, $T\in X$.

1. (1) $D(\sigma, K):=\{\langle \sigma _0, \sigma _1,\dots, \sigma _p\rangle \in K' \mid \sigma _p\geq \sigma \}$

2. (2) $D(\sigma, \tau ):=\{\langle \sigma _0, \sigma _1,\dots, \sigma _p\rangle \in K' \mid \sigma _p\geq \sigma, \; \tau \geq \sigma _0 \}$, the dual cell of $\sigma$ in $\tau$.

3. (3) $D_\sigma T:=\{\langle S_0, S_1,\dots, S_p\rangle \in X' \mid \sigma \leq \pi (S_p),\; S_0\leq T\}$

4. (4) $T_\sigma := |D_\sigma T|$. (Therefore, $T_\sigma =(\pi \mid T)^{-1}|D(\sigma,\pi (T))|$).

Of course, $D(\sigma, \tau )=\emptyset$ unless $\sigma \leq \tau$, and $D_\sigma T=\emptyset$ unless $\sigma \leq \pi (T)$.

$D_\sigma T$ is a subcomplex of $X'$. $D(\sigma, K)$ and $D(\sigma, \tau )$ are subcomplexes of $K'$.

Lemma 7.4 Let $(X,\pi )$ be a $K$-space. Suppose $\sigma \in K$, $T\in X$, and $\sigma \leq \pi (T)$.

1. (1) $T_\sigma =|D_\sigma T|$ is a pl ball. $dim(T_\sigma ) = dim(T)-dim(\sigma )$.

2. (2) $\partial D_\sigma T=\partial ^i D_\sigma T \cup \partial ^o D_\sigma T$, (the inner and outer boundaries) where:

   \[ \partial^i D_\sigma T={\cup} \{D_\rho T\;\mid \sigma< \rho \};\quad \partial^oD_\sigma T={\cup} \{D_\sigma S\;\mid S< T\} \]

3. (3) Suppose $\sigma <\pi (T)$. Then $|\partial ^iD_\sigma T|$ and $|\partial ^o D_\sigma T|$ are pl balls of dimension $dim(D_\sigma T)-1$, and

   \[ \partial (\partial^iD_\sigma T)=\partial(\partial^oD_\sigma T)=\partial^iD_\sigma T\cap \partial^oD_\sigma T\;. \]

Proof. of (1): For each vertex $v$ of $\tau$ note that,

\[ |D(v,\tau)|=\{x\in \tau \;| \;a_v(x)\geq a_w(x),\text{ for all vertices}\ w\ {\rm of\ }\tau\}. \]

where $a_v: |K|\to [0,1]$ denotes the barycentric coordinate function defined by the vertex $v$. This is a convex subset of $\tau$. So

\[ |D(\sigma, \tau)|=\cap_{v\in V(K)} |D(v, \tau)| \]

is also convex. Therefore $T_\sigma =(\pi _{|T})^{-1} (|D(\sigma, \tau )|)$ is also convex since $\pi _{|T}: T\to \tau$ is simplicial. So $T_\sigma$ is a compact convex polyhedron and therefore a pl ball.

Since $|D(\sigma, \tau )|\cap \tau ^\circ \neq \emptyset$, this operator $(\pi _{|T})^{-1}$ preserves codimension:

\[ dim(\tau) - dim(D(\sigma, \tau)) = dim (T) - dim (D_\sigma T). \]

Since $dim(D(\sigma, \tau )) = dim(\tau )- dim(\sigma )$, we get: $dim(D_\sigma T)= dim(T)-dim(\sigma )$.

Proof. of (2): See [3], proposition 5.6(2), applied to $\pi _{|\overline {T}} :\overline {T}\to \pi (\overline {T})$.

Proof. of (3): The equation in (3), and the fact that $|\partial ^iD_\sigma T|$ and $|\partial ^o D_\sigma T|$ are both pl manifolds, are proved in [3] [proposition 5.6 (3),(4)]. To show $|\partial ^iD_\sigma T|$ is a pl ball, it suffices to note that it collapses to the vertex $bT$, and so $|\partial ^iD_\sigma T|$ is a regular neighbourhood of $bT$ in $|\partial D_\sigma T|$ (by 3.30 of [14]). Then by 3.13 of [14], $\partial ^oD_\sigma T$ is also a pl ball.

Definition 7.5 ([14] p.27) A ball complex is a finite collection $Z=\{B_i\}_{i\in I}$ of pl balls in a euclidean space, such that each point of $|Z|:=\cup \{ B\;\mid \; B\in Z\}$ lies in the interior of precisely one ball of $Z$, and the boundary of each $B\in Z$ is a union of balls of lesser dimension of $Z$. Therefore $(|Z|, Z)$ is a regular CW-complex.

Let $Z$ and $Y$ be ball complexes A pl map $f:|Z|\to |Y|$ is a map of ball complexes if for each ball $B$ of $Z$, $f(B)$ is a ball of $Y$.

Definition 7.6 Let $(X,\pi )$ be a $K$-space. We define

\[ X_K = \{T_\sigma \;\mid \sigma\in K, \; T\in X, \; \sigma\leq \pi(T)\} \]

Theorem 7.7 Let $(X,\pi )$ be a $K$-space. Then $X_K$ is a ball complex. Moreover $X'$ is a simplicial subdivision of $X_K$. Also, $X_K$ is a subdivision of $X$.

Let $f:(X,\pi _X)\to (Y,\pi _Y)$ is a map of $K$-spaces. The induced map $f':X'\to Y'$ of derived complexes is then a map of ball complexes, $f_K: X_K\to Y_K$.

Proof. (The induced map $f'$ means the simplicial map $f':X'\to Y'$ for which $f'(bS)= b(f(S))$ for each $S\in X.$) By lemma 7.4 the boundary of each $T_\sigma$ is a union of balls of $X_K$ with smaller dimension and

\[ T_\sigma ^\circ{=} \coprod \{A^\circ\;\mid A=\langle S_0, \ldots, S_p\rangle\in D_\sigma T, \; A\notin \partial^iD_\sigma T, \; A\notin \partial^oD_\sigma T \} . \]

This can be rewritten as:

(7.1)\begin{equation} T_\sigma^\circ{=} \coprod\{A^\circ\;\mid A=\langle S_0, \ldots, S_p\rangle\in X', \; \sigma=\pi(S_p), \; T= S_0\}, \end{equation}

By equation (7.1), for each $A\in X'$ there is a unique $T_\sigma \in X_K$ for which $A^\circ \subset T_\sigma ^\circ$. Therefore : $|X'|= \coprod \{T_\sigma ^\circ \; \mid \; \; T_\sigma \in X_K \}=|X_K|$.

This proves that $X_K$ is a ball complex and that $X'$ is a subdivision of $X_K$. Because $T_\sigma \subset T$, we see $X_K$ is a subdivision of $X$.

Now let $f:(X,\pi _X)\to (Y, \pi _Y)$ be a map of $K$-spaces. For each simplex $S\in X$ we see $f(S)\in Y$ because $f$ is simplicial. For each face $\sigma$ of $\pi _X(S)$ in $K$, we see from the definitions that $f'(D_\sigma S)= D_\sigma f(S)$. So $f'$ is a map of ball complexes, $f_K: X_K \to Y_K$.

8. The isomorphism $\Phi _X:T\Delta ^*X\cong C(X_K)$

Our main theorem is:

Theorem 8.1 For each $K$-space $(X, \pi )$ the cellular chain complex of $X_K$ with $R$ coefficients, denoted $C(X_K)$, comes with a natural $(R,K)$complex structure. There is defined (below) an isomorphism of $(R,K)$ chain complexes:

\[ \Phi_X :T\Delta^*X\cong C(X_K)\;. \]

For each map $f:(X, \pi _X)\to (Y,\pi _Y)$ of $K$-spaces, the square below commutes.

Proof. Choose a basis $bK$ of oriented cells for $\Delta _*(K;R)$. Choose next, a basis $b_*X$ of oriented cells for $\Delta _*(X;R)$. But choose the orientations in $b_*X$ so that if $T\in b_*X$ and $\sigma \in bK$ are both $q$-cells, and if $\pi _*(T)=\pm \sigma \in \Delta _q(K;R)$, then:

\[ \pi_*(T) =({-}1)^{dim(\sigma)}\sigma \in \Delta_q(K;R). \]

We call such a pair, $(bK, b_*X)$ an orientation for $(X,\pi )$.

Our first task is to construct the cellular chain complex $C_*(X_K;R)$ as the underlying $R$-complex of an $(R,K)$ complex $C(X_K)$. Define

\[ C(X_K)= \Delta X\otimes_K\Delta^*K; \quad C_*(X_K;R)=(\Delta X\otimes_K\Delta^*K)(K) \]

For each oriented simplex $\rho \in bK$ and oriented simplex $T\in b_*X$, define

\[ [T_\rho] = T\otimes_K \rho^*\in C_{|T|-|\sigma|}(X_K;R)\quad(\text{where}\ |\sigma|=dim(\sigma)). \]

(The geometric intuition for this definition is the fact that, the map $C_X$, of corollary 9.3, takes $T\otimes _K \rho ^*$ to a fundamental cycle, in $\Delta X'$, for the cell $D_\rho T$, whose underlying space is $T_\rho$).

Define $bX_K= \{ [T_\rho ]\mid T\in b_*X,\; \rho \in bK, \;T_\rho \leq \pi (T)\}$. Then $bX_K$ is an $R$-basis for $C_*(X_K;R)$ in bicorrespondence with the cells of $X_K$. Write $\partial _q$ for the boundary map in $C_*(X_K;R)$, namely: $\partial _q=(d^{\Delta X\otimes _K\Delta ^*K})_q$.

But to justify these definitions, we must check that $C_*(X_K;R)$ does compute the cellular homology of $X_K$. It suffices to check, for any $[T_\rho ]\in bX_K,$ that $\partial _q([T_\rho ])$ is a sum with $\pm 1$ coefficients of those $[S_\sigma ]\in bX_K$ which are $(q-1)$-faces of $T_\rho$. (See [7], for example.)

All proper faces of $T_\rho$ have the form $T_\sigma,$ for $\rho <\sigma$, or $S_\rho$, for $S< T$.

Suppose $[T_\rho ]\in bX_K$. So $T\in b_*X$, $\rho \in bK$. Set $\tau = \pi (T)\in K$. By (5.1):

\begin{align*} \partial_q[T_\rho]& = d^{\Delta X\otimes_K\Delta^*K} (T\otimes_K\rho^*) \\ & =\sum_{\{\sigma\mid \rho\leq \sigma\leq \tau\}} \{(d^{\Delta X} (\sigma,\tau)T)\otimes\rho^*+({-}1)^{|T|} T\otimes d^{\Delta^*K}(\sigma, \rho)\rho^*\}\\ & = \sum_{S< T}[T,S][S_\rho] +({-}1)^{1+|T_\rho|} \sum_{\rho<\sigma}[\sigma,\rho][T_\sigma] \end{align*}

which is as required.

This completes the construction of the cellular chain complex of $X_K$, as an $(R,K)$ complex, $C(X_K)$.

The $(R,K)$ isomorphism, $\Phi _X: T\Delta ^* X\cong C(X_K)$ is simply:

\[ \Phi_X:= (\varepsilon_{\Delta X}\otimes_K 1_{\Delta^*K}):T\Delta^*X=\Delta^{**}X \otimes_K\Delta^*K\longrightarrow \Delta X\otimes_K\Delta^*K= C(X_K). \]

Naturality of $\Phi$ is obvious from the naturality of $\varepsilon$.

9. The McCrory cap product, $\Delta ^*X$ and $\Delta X'$

We now use the work of McCrory [13] to construct, for any $K$-space, $(X, \pi )$, an $(R,K)$ chain monomorphism $C(X_K)\mathop {\longrightarrow }\limits ^{C_X} \Delta X'.$ serving two purposes.

First, it defines an $(R,K)$ chain homotopy equivalence, $T\Delta ^*X\simeq \Delta X'$.

Second, $C_X$ identifies $C(X_K)$ with that $(R, K)$ subcomplex of $\Delta X'$ which admits a basis consisting of one fundamental $q$-cycle, in $\Delta _q (D_\sigma T,\partial D_\sigma T)\subset \Delta _q(X')$, for each $q$-cell $T_\sigma$ of $X_K$. (This will complete our geometric interpretation of $T$).

Let $K$ be a finite simplicial complex. McCrory (see [13], and also [11]) defines a map, $c': \Delta _*(K;R)\otimes _R\Delta ^*(K;R)\to \Delta _*(K';R)$ which he shows is chain homotopic to the composite,

\[ \Delta_*(K;R)\otimes_R\Delta^*(K;R)\mathop{\longrightarrow}\limits^{{\cap}} \Delta_*(K;R)\mathop{\longrightarrow}\limits^{Sd} \Delta_*(K';R) \]

where $\cap$ denotes the Whitney-Cech cap product. We will write $c_K$ for $c'$. We repeat his definition here with appropriate sign changes because McCrory's sign conventions differ slightly from ours.

For any $q$-simplex, $Q=\langle Q^0, Q^1, \ldots Q^q\rangle \text { of } K'$ in which each $Q_i$ is oriented, McCrory then defines

\[ \varepsilon(Q) = [Q^0,Q^1][Q^1,Q^2]\ldots [Q^{q-1},Q^q]. \]

This is independent of the orientations on $Q_1, Q_2,\ldots Q_{q-1}$. If $q=0$, set $\varepsilon (Q)=0$.

For any $n$-simplex $\tau$ and $(n-q)$-simplex $\sigma$ of $K$, each simplex $Q=\langle Q^0, Q^1, \ldots Q^q\rangle$ of $D(\sigma, \tau )_q$ satisfies: $Q^0= \tau ; \;Q^q=\sigma$. Therefore, $\varepsilon (Q)$ makes sense if $\tau$ and $\sigma$ are oriented simplices chosen from some basis $bK$ of oriented simplices for $\Delta _*(K;R)$ (but not if $\sigma =-\tau ).$

The McCrory Cap Product, $\Delta _*(K;R)\otimes _R\Delta ^*(K;R)\mathop {\longrightarrow }\limits ^{c_K} \Delta _*(K';R)$ is the map defined by:

\[ \boxed{ c_K(\tau\otimes \sigma^*)= \sum_{Q\in D(\sigma,\tau)_q}({-}1)^{dim (\sigma)} \varepsilon(Q)Q } \]

for any oriented simplices $\sigma, \tau$ in some basis $bK$. Here $q= dim(\tau )-dim(\sigma )$. Note this is zero unless $\sigma \leq \tau$. Note that $c_K$ does not change if we change the basis.

$c_K$ is a chain map. We reprove this in Appendix I, § A, because of the sign changes and because McCrory's proof, [13] p.155 lines 7-8, is only a sketch.

Now suppose $(X,\pi )$ is a $K$-space.

Note that if $T$ and $\sigma$ are oriented simplices of $X$ and $K$ and $q=dim(T)-dim(\sigma )\neq 0$:

\[ c_X(T\otimes_R \pi^*\sigma^*)=\sum_{Q\in (D_\sigma, T)_q}({-}1)^{dim (\sigma)} \varepsilon(Q)Q\ \in \Delta_qX'(\sigma) \]

(because $D_\sigma T=\cup \{D(S,T)\; |\; S\in X, dim(S)=dim(\sigma ), \pi (S)= \sigma \}$). This formula still makes sense and is true if $q=0$ and $\pi _*(T)\neq -\sigma$).

In this way, $c_X\circ (1\otimes \pi ^*)$ defines an $(R,K)$ chain map,

\[ c_X\circ(1\otimes \pi^*): \Delta X\otimes_R\Delta^*K\longrightarrow \Delta X' \]

Proposition 9.1 There is a unique $(R,K)$ chain map

\[ C_X: C(X_K)=\Delta X\otimes_K \Delta^*K \longrightarrow \Delta X' \]

satisfying:

\[ c_X\circ(1\otimes \pi^*)=C_X\circ \pi_{\Delta X, \Delta^*K} \]

$C_X$ is an $(R,K)$ monomorphism. For all $q$-cells $T_\sigma$ of $X_K$, with $q\neq 0$,

\[ C_X(T\otimes_K \sigma^*)= \sum_{Q\in (D_\sigma T)_q}({-}1)^{dim (\sigma)} \varepsilon(Q)Q. \]

For a $0$-cell $T_\sigma$, of $X_K$, with $T\in \Delta _n(X;R), \sigma \in \Delta _n(K;R)$ oriented so that $\pi _*(T)=\sigma$, then

\[ C_X(T\otimes_K\sigma^*)= ({-}1)^{dim(T)}\langle T \rangle, \quad (\langle T\rangle \hbox{ is the barycenter } bT \hbox{ of } T). \]

Proof. Note that $c_X(T\otimes _R \pi ^*\sigma ^*)=0$ unless $\pi (T)\geq \sigma$. Also $c_X(T\otimes _R \pi ^*\sigma ^*)\in \Delta X'(\sigma )$ for all $\sigma \in K$ and $T\in X$ because each $q$-cell $Q\in D_\sigma T$ is in $\Delta _qX'(\sigma )$ if $q= dim(T_\sigma )$.

So $c_X\,\circ (1\otimes \pi ^*):\Delta X\otimes _R \Delta ^*K\to \Delta X'$ is an $(R,K)$ chain map annihilating each $\Delta X(K-st(\sigma ))\otimes _R \Delta ^*K(\sigma )$. Hence there is a unique $(R,K)$ chain map monomorphism, $\Delta X\otimes _K \Delta ^*K\mathop {\longrightarrow }\limits ^{C_X} \Delta X'$ such that $c_X\circ (1\otimes \pi ^*) = C_X\circ \pi _{\Delta X, \Delta ^*K}$. The calculation follows if $q\neq 0$. If $q=0$, then $(\pi _{\mid T})^* \sigma ^*=T^*$, so

\[ C_X(T\otimes_K \sigma^*)= c_X(T\otimes T^*) =({-}1)^{dim (T)}\sum_{Q\in D(T,T)_0}Q= ({-}1)^{dim(T)}\langle T\rangle \]

Clearly $C_X$ is natural in $(X,\pi )$.

Remark 9.2 If we choose an orientation $(bK, b_*X)$ for $X_K$, then for each $0$-cell $T_\sigma = T_{\pi (T)}$ of $X_K$, with $[T_\sigma ]\in bX_K$, we have $C_X([T_\sigma ])= \langle T\rangle \in \Delta _0(X';R)$.

Corollary 9.3 For each $q$-cell $T_\sigma$ of $X_K$, $C_X(T\otimes \sigma ^*)$ is a fundamental cycle, in $\Delta _q(D_\sigma T, \partial D_\sigma T; R)$ for the $q$-manifold $D_\sigma T$.

Proof. $C_X(T\otimes _K \sigma ^*)$ is a fundamental cycle in $\Delta _q(D_\sigma T, \partial D_\sigma T; R)$ since $C_X$ is a chain map and since each $Q\in (D_\sigma T)_q$ appears with coefficient $\pm 1$ in $C_X(T\otimes _K \sigma ^*)$.

Theorem 9.4 For each $K$-space $(X,\pi )$, the map $C(X_K)\mathop {\longrightarrow }\limits ^{C_X} \Delta X'$ is an $(R,K)$ chain homotopy equivalence.

Proof. By 9.3, for all $T_\sigma$, $C_X$ restricts to a homotopy equivalence,

\[ C_*(T_\sigma,\partial T_\sigma; R)\to \Delta_*(D_\sigma(T),\partial D_\sigma(T);R) \]

and it takes chains on any subcomplex of $X_K$ to chains on its subdivision. By an induction-excision argument on the number of cells in the subcomplex one sees $C_X$ yields a homology equivalence and then a chain homotopy equivalence on each such subcomplex. So $C_X(\sigma,\sigma )$ is an $R$-chain equivalence for each $\sigma$. Therefore $C_X$ is an $(R,K)$ chain equivalence.

Together, 9.4 and 8.1 clearly prove:

Corollary 9.5 $T\Delta ^*X\mathop {\longrightarrow }\limits ^{ C_X \Phi _X} \Delta X'$ is an $(R,K)$ chain homotopy equivalence. Consequently $e_{\Delta ^*X}\circ T(C_X\Phi _X)$ is an explicit $(R,K)$ chain homotopy equivalence,

\[ T\Delta X'\simeq \Delta^*X. \]