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Weighted projective spaces and iterated Thom spaces. (English) Zbl 1290.57030

From the summary: “For any weight vector \(\chi\) of positive integers, the weighted projective space \(\mathbf{P}(\chi)\) 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 \(\mathbf{P}(\chi)\), paying particular attention to its localisation at individual primes \(p\). We identify certain \(p\)-primary weight vectors \(\pi\) for which \(\mathbf{P}(\pi)\) 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 \(\mathbf{P}(\chi)\), 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 \(\mathbf{P}(\chi)\), and may be interpreted as a resolution of singularities.”

MSC:

57R18 Topology and geometry of orbifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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