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Weighted projective spaces and iterated Thom spaces | | Anthony Bahri
; Matthias Franz
; Nigel Ray
; | | Date: |
12 Sep 2011 | | 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. | | Source: | arXiv, 1109.2359 | | Services: | Forum | Review | PDF | Favorites | |
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