A canonical subspace of $H^ *(B\textrm {O})$ and its application to bordism
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- by Errol Pomerance PDF
- Trans. Amer. Math. Soc. 309 (1988), 659-670 Request permission
Abstract:
A particularly nice canonical subspace of ${H^{\ast }}(BO)$ is defined. The bordism class of a map $f:X \to Y$, where $X$ and $Y$ are compact, closed manifolds, can be determined by the characteristic numbers corresponding to elements of this subspace, and these numbers can be easily calculated. As an application, we study the "fixed-point manifold" of a parameter family of self-maps $F:M \times X \to X$, thus refining to bordism the usual homological analysis of the diagonal which is the basis of the standard Lefschetz fixed point theorem.References
- M. F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200–208. MR 126856, DOI 10.1017/s0305004100035064
- Edgar H. Brown Jr. and Franklin P. Peterson, Relations among characteristic classes. I, Topology 3 (1964), no. suppl, suppl. 1, 39–52. MR 163326, DOI 10.1016/0040-9383(64)90004-7
- Edgar H. Brown Jr. and Franklin P. Peterson, Some remarks about symmetric functions, Proc. Amer. Math. Soc. 60 (1976), 349–352 (1977). MR 433465, DOI 10.1090/S0002-9939-1976-0433465-6
- Shaun Bullett, Permutations and braids in cobordism theory, Proc. London Math. Soc. (3) 38 (1979), no. 3, 517–531. MR 532985, DOI 10.1112/plms/s3-38.3.517
- F. R. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), no. 2, 99–110. MR 493954, DOI 10.1007/BF01393249
- P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
- D. B. Fuks, Cohomology of the braid group $\textrm {mod}\ 2$, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 62–73 (Russian). MR 0274463
- Mark Mahowald, A new infinite family in ${}_{2}\pi _{*}{}^s$, Topology 16 (1977), no. 3, 249–256. MR 445498, DOI 10.1016/0040-9383(77)90005-2
- Mark Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979), no. 3, 549–559. MR 544245
- John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
- Stewart Priddy, $K(\textbf {Z}/2)$ as a Thom spectrum, Proc. Amer. Math. Soc. 70 (1978), no. 2, 207–208. MR 474271, DOI 10.1090/S0002-9939-1978-0474271-8 K. Reidemeister, Knotentheorie, Ergeb. Math. Grenzgeb. (Alte Folge), Band 1, Heft 1, Springer, Berlin, 1932; English transl., B. C. S. Assoc., Moscow, Idaho, 1983.
- Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
- René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 309 (1988), 659-670
- MSC: Primary 57R75
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961606-3
- MathSciNet review: 961606