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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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References:

[1] J.F. Adams: Stable Homotopy and Generalised Homology, Univ. Chicago Press, Chicago, IL, 1974. · Zbl 0309.55016
[2] A. Al Amrani: Complex \(K\)-theory of weighted projective spaces , J. Pure Appl. Algebra 93 (1994), 113-127. · Zbl 0808.55004 · doi:10.1016/0022-4049(94)90106-6
[3] A. Al Amrani: Cohomological study of weighted projective spaces ; in Algebraic Geometry (Ankara, 1995), Lecture Notes in Pure and Appl. Math. 193 , Dekker, New York, 1997, 1-52. · Zbl 0899.14027
[4] A. Al Amrani: A comparison between cohomology and \(K\)-theory of weighted projective spaces , J. Pure Appl. Algebra 93 (1994), 129-134. · Zbl 0808.55005 · doi:10.1016/0022-4049(94)90107-4
[5] A. Bahri, M. Franz and N. Ray: The equivariant cohomology ring of weighted projective space , Math. Proc. Cambridge Philos. Soc. 146 (2009), 395-405. · Zbl 1205.14022 · doi:10.1017/S0305004108001965
[6] A. Bahri, M. Franz, D. Notbohm and N. Ray: The classification of weighted projective spaces , Fund. Math. 220 (2013), 217-226. · Zbl 1271.55006 · doi:10.4064/fm220-3-3
[7] Y. Civan and N. Ray: Homotopy decompositions and \(K\)-theory of Bott towers , \(K\)-Theory 34 (2005), 1-33. · Zbl 1078.55016 · doi:10.1007/s10977-005-1551-x
[8] M.W. Davis and T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions , Duke Math. J. 62 (1991), 417-451. · Zbl 0733.52006 · doi:10.1215/S0012-7094-91-06217-4
[9] I. Dolgachev: Weighted projective varieties ; in Group Actions and Vector Fields (Vancouver, B.C., 1981), Lecture Notes in Math. 956 , Springer, Berlin, 1982, 34-71. · Zbl 0516.14014 · doi:10.1007/BFb0101508
[10] W. Fulton: Introduction to Toric Varieties, Annals of Mathematics Studies 131 , Princeton Univ. Press, Princeton, NJ, 1993. · Zbl 0813.14039
[11] M. Grossberg and Y. Karshon: Bott towers, complete integrability, and the extended character of representations , Duke Math. J. 76 (1994), 23-58. · Zbl 0826.22018 · doi:10.1215/S0012-7094-94-07602-3
[12] M. Hazewinkel: Formal Groups and Applications, Academic Press, New York, 1978. · Zbl 0454.14020
[13] A.R. Iano-Fletcher: Working with weighted complete intersections ; in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281 , Cambridge Univ. Press, Cambridge, 2000, 101-173. · Zbl 0960.14027 · doi:10.1017/CBO9780511758942.005
[14] T. Kawasaki: Cohomology of twisted projective spaces and lens complexes , Math. Ann. 206 (1973), 243-248. · Zbl 0268.57005 · doi:10.1007/BF01429212
[15] Y. Nishimura and Z. Yosimura: The quasi \(K\mathrm{O}_{*}\)-types of weighted projective spaces , J. Math. Kyoto Univ. 37 (1997), 251-259. · Zbl 0910.55002
[16] D.C. Ravenel: Complex Cobordism and Stable Homotopy Groups of Spheres, second edition, AMS Chelsea, 2003.
[17] N. Ray: On a construction in bordism theory , Proc. Edinburgh Math. Soc. (2) 29 (1986), 413-422. · Zbl 0603.57021 · doi:10.1017/S0013091500017855
[18] R.H. Szczarba: On tangent bundles of fibre spaces and quotient spaces , Amer. J. Math. 86 (1964), 685-697. · Zbl 0151.31703 · doi:10.2307/2373152
[19] V. Welker, G.M. Ziegler and R.T. Živaljević: Homotopy colimits–comparison lemmas for combinatorial applications , J. Reine Angew. Math. 509 (1999), 117-149. · Zbl 0995.55004 · doi:10.1515/crll.1999.035
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