A Homotopy Theory of Orbispaces

Abstract

In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist's Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related to the so called McKay correspondence in mathematics (cf. [McK]). Later developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of Calabi-Yau orbifolds (cf. [Ro]), and most recently the Gromov-Witten invariants of symplectic orbifolds (cf. [CR1-2]). One common feature of these studies is that certain contributions from singularities, which are called ``twisted sectors'' in physics, have to be properly incorporated. This is called the ``stringy aspect'' of an orbifold (cf. [R]). This paper makes an effort to understand the stringy aspect of orbifolds in the realm of ``traditional mathematics''.

latex, 59 pages, minor mistakes corrected, more references added

References in corpus (5)

• A New Cohomology Theory for Orbifold
• String Topology
• Topological Orbifold Models and Quantum Cohomology Rings
• McKay correspondence
• Orbifold Quantum Cohomology

Cited by in corpus (12)

• Lie Groupoids and Differentiable Stacks
• Orbispaces as differentiable stratified spaces
• Iteration of Cox rings of klt singularities
• Orbifolds and stable homotopy groups
• Proper Orbifold Cohomology
• The fundamental group of symplectic manifolds with Hamiltonian Lie group actions
• On the 1-homotopy type of Lie groupoids
• The fundamental group of quotients of products of some topological spaces by a finite group -- A generalization of a Theorem of Bauer-Catanese-Grunewald-Pignatelli
• Closed geodesics on orbifolds
• The isomorphism conjecture for Artin groups
• Gerbes and twisted orbifold quantum cohomology
• On a notion of maps between orbifolds, I. function spaces