A Homotopy Theory of Orbispaces

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''.