Weighted projective spaces and iterated Thom spaces

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.