Weighted projective spaces and iterated Thom spaces.

Bahri, Anthony and Franz, Matthias and Ray, Nigel (2011) Weighted projective spaces and iterated Thom spaces. [MIMS Preprint]

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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, 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.

Item Type:	MIMS Preprint
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topology MSC 2010, the AMS's Mathematics Subject Classification > 57 Manifolds and cell complexes
Depositing User:	Nigel Ray
Date Deposited:	06 Sep 2011
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1668

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