Weighted projective spaces and iterated Thom spaces

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

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