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January 2014 Weighted projective spaces and iterated Thom spaces
Anthony Bahri, Matthias Franz, Nigel Ray
Osaka J. Math. 51(1): 89-121 (January 2014).


For any weight vector $\chi$ of positive integers, the weighted projective space $\mathbb{P}(\chi)$ is a projective toric variety, and has orbifold singularities in every case other than standard projective space. Our principal aim is to study the algebraic topology of $\mathbb{P}(\chi)$, paying particular attention to its localisation at individual primes $p$. We identify certain $p$-primary weight vectors $\pi$ for which $\mathbb{P}(\pi)$ is homeomorphic to an iterated Thom space, and discuss how any weighted projective space may be reassembled from its $p$-primary parts. The resulting Thom isomorphisms provide an alternative to Kawasaki's calculation of the cohomology ring of $\mathbb{P}(\chi)$, and allow us to recover Al Amrani's extension to complex $K$-theory. Our methods generalise to arbitrary complex oriented cohomology algebras and their dual homology coalgebras, as we demonstrate for complex cobordism theory, the universal example. In particular, we describe a fundamental class that belongs to the complex bordism coalgebra of $\mathbb{P}(\chi)$, and may be interpreted as a resolution of singularities.


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Anthony Bahri. Matthias Franz. Nigel Ray. "Weighted projective spaces and iterated Thom spaces." Osaka J. Math. 51 (1) 89 - 121, January 2014.


Published: January 2014
First available in Project Euclid: 8 April 2014

zbMATH: 1290.57030
MathSciNet: MR3192533

Primary: 57R18
Secondary: 14M25 , 55N22

Rights: Copyright © 2014 Osaka University and Osaka City University, Departments of Mathematics


Vol.51 • No. 1 • January 2014
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