Weighted projective spaces and iterated Thom spaces.

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, which is the universal example. In particular, we describe a fundamental class in Omega_{2n}^U(P(\chi)), which may be interpreted as a resolution of singularities