The complex bordism of groups with periodic cohomology
HTML articles powered by AMS MathViewer
- by Anthony Bahri, Martin Bendersky, Donald M. Davis and Peter B. Gilkey
- Trans. Amer. Math. Soc. 316 (1989), 673-687
- DOI: https://doi.org/10.1090/S0002-9947-1989-0942423-8
- PDF | Request permission
Abstract:
Is is proved that if $BG$ is the classifying space of a group $G$ with periodic cohomology, then the complex bordism groups $M{U_{\ast }}(BG)$ are obtained from the connective $K$-theory groups $k{u_{\ast }}(BG)$ by just tensoring up with the generators of $M{U_{\ast }}$ as a polynomial algebra over $k{u_{\ast }}$. The explicit abelian group structure is also given. The bulk of the work is the verification when $G$ is a generalized quaternionic group.References
- J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716
- M. F. Atiyah, $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR 0224083
- Martin Bendersky and Donald M. Davis, On the complex bordism of classifying spaces, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 53–56. MR 1000366, DOI 10.1007/BFb0085217
- Edgar H. Brown Jr. and Franklin P. Peterson, A spectrum whose $Z_{p}$ cohomology is the algebra of reduced $p^{th}$ powers, Topology 5 (1966), 149–154. MR 192494, DOI 10.1016/0040-9383(66)90015-2
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Fred Cohen, Splitting certain suspensions via self-maps, Illinois J. Math. 20 (1976), no. 2, 336–347. MR 405412
- P. E. Conner and E. E. Floyd, Periodic maps which preserve a complex structure, Bull. Amer. Math. Soc. 70 (1964), 574–579. MR 164356, DOI 10.1090/S0002-9904-1964-11204-0 —, The relation of cobordism to $K$-theories, Lecture Notes in Math., vol. 28, Springer-Verlag, Berlin, 1966.
- P. E. Conner and Larry Smith, On the complex bordism of finite complexes, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 117–221. MR 267571
- Donald M. Davis, The BP-coaction for projective spaces, Canadian J. Math. 30 (1978), no. 1, 45–53. MR 478151, DOI 10.4153/CJM-1978-004-9
- Kensô Fujii and Masahiro Sugawara, The additive structure of $\~K(S^{4n+3}/Q_{t})$, Hiroshima Math. J. 13 (1983), no. 3, 507–521. MR 725962
- Peter B. Gilkey, The eta invariant and the equivariant unitary bordism of spherical space form groups, Compositio Math. 65 (1988), no. 1, 33–50. MR 930146
- Shin Hashimoto, On the connective $K$-homology groups of the classifying spaces $B\textbf {Z}/p^{r}$, Publ. Res. Inst. Math. Sci. 19 (1983), no. 2, 765–771. MR 716975, DOI 10.2977/prims/1195182451
- Dale Husemoller, Fibre bundles, 2nd ed., Graduate Texts in Mathematics, No. 20, Springer-Verlag, New York-Heidelberg, 1975. MR 0370578
- David Copeland Johnson, A Stong-Hattori spectral sequence, Trans. Amer. Math. Soc. 179 (1973), 211–225. MR 368040, DOI 10.1090/S0002-9947-1973-0368040-7
- Teiichi Kobayashi and Masahiro Sugawara, Note on $\textrm {KO}$-rings of lens spaces mod $2^{r}$, Hiroshima Math. J. 8 (1978), no. 1, 85–90. MR 485765
- Abdeslam Mesnaoui, Unitary bordism of classifying spaces of quaternion groups, Pacific J. Math. 142 (1990), no. 1, 49–67. MR 1038729
- Stephen A. Mitchell and Stewart B. Priddy, Symmetric product spectra and splittings of classifying spaces, Amer. J. Math. 106 (1984), no. 1, 219–232. MR 729761, DOI 10.2307/2374436
- Richard G. Swan, Groups with periodic cohomology, Bull. Amer. Math. Soc. 65 (1959), 368–370. MR 115175, DOI 10.1090/S0002-9904-1959-10378-5
- Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
Bibliographic Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 316 (1989), 673-687
- MSC: Primary 55N22
- DOI: https://doi.org/10.1090/S0002-9947-1989-0942423-8
- MathSciNet review: 942423