A Homotopy Theory of Orbispaces

An orbifold is a singular space which is locally modeled on the quotient of a smooth manifold by a smooth action of a finite group. It appears naturally in geometry and topology when group actions on manifolds are involved and the stabilizer of each fixed point is finite. The concept of an orbifold was first introduced by Satake under the name “V -manifold” in a paper where he also extended the basic differential geometry to his newly defined singular spaces (cf. [Sa]). The local structure of an orbifold – being the quotient of a smooth manifold by a finite group action – was merely used as some “generalized smooth structure”. A different aspect of the local structure was later recognized by Thurston, who gave the name “orbifold” and introduced an important concept – the fundamental group of an orbifold (cf. [Th]). 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”. Surprisingly, we were led to a refinement of Thurston’s discovery!