Weighted projective spaces and iterated Thom spaces

TL;DR

This paper explores the algebraic topology of weighted projective spaces, identifying conditions under which they resemble iterated Thom spaces, and extends cohomology computations to complex oriented theories.

Contribution

It characterizes weighted projective spaces as iterated Thom spaces for certain primes and generalizes cohomology computations to complex oriented theories.

Findings

Identification of p-primary weight vectors with iterated Thom space structure

Expression of Kawasaki's cohomology computations via Thom isomorphisms

Extension of methods to complex cobordism and other complex oriented theories

Abstract

For any (n+1)-dimensional weight vector {\chi} of positive integers, the weighted projective space P(\chi) is a projective toric variety, and has orbifold singularities in every case other than CP^n. We study the algebraic topology of P(\chi), paying particular attention to its localisation at individual primes p. We identify certain p-primary weight vectors {\pi} for which P(\pi) is homeomorphic to an iterated Thom space over S^2, and discuss how any P(\chi) may be reconstructed from its p-primary factors. We express Kawasaki's computations of the integral cohomology ring H^*(P(\chi);Z) in terms of iterated Thom isomorphisms, and recover Al Amrani's extension to complex K-theory. Our methods generalise to arbitrary complex oriented cohomology algebras E^*(P(\chi)) and their dual homology coalgebras E_*(P(\chi)), as we demonstrate for complex cobordism theory (the universal example). In particular, we describe a fundamental class in \Omega^U_{2n}(P(\chi)), which may be interpreted as a resolution of singularities.