We show that every orbispace satisfying certain mild hypotheses has ‘enough’ vector bundles. It follows that the $K$-theory of finite rank vector bundles on such orbispaces is a cohomology theory. Global presentation results for smooth orbifolds and derived smooth orbifolds also follow.

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specializing to half-dimensional torus actions.

For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

An invertible polynomial in n variables is a quasi-homogeneous polynomial consisting of n monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau–Ginzburg models, Berglund, Hübsch and Henningson considered a pair (f, G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair . Here we study the reduced orbifold zeta functions of dual pairs (f, G) and and show that they either coincide or are inverse to each other depending on the number n of variables.

The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one, which are both Seifert fibred.

The paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.