The equivariant Conner-Floyd isomorphism

Costenoble, Steven R.

doi:10.1090/S0002-9947-1987-0911096-0

The equivariant Conner-Floyd isomorphism
HTML articles powered by AMS MathViewer

by Steven R. Costenoble PDF

Trans. Amer. Math. Soc. 304 (1987), 801-818 Request permission

Abstract:

This paper proves two equivariant generalizations of the Conner-Floyd isomorphism relating unitary cobordism and $K$-theory. It extends a previous result of Okonek for abelian groups to all compact Lie groups. We also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism.

References

Lectures on Lie groups

M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. (2) 19 (1968), 113–140. MR 228000, DOI 10.1093/qmath/19.1.113
M. F. Atiyah and G. B. Segal, Equivariant $K$-theory and completion, J. Differential Geometry 3 (1969), 1–18. MR 259946
Theodor Bröcker and Edward C. Hook, Stable equivariant bordism, Math. Z. 129 (1972), 269–277. MR 321119, DOI 10.1007/BF01187353
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

The relation of cobordism to

theories

Equivariant cobordism and

theory

Tammo tom Dieck, Bordism of $G$-manifolds and integrality theorems, Topology 9 (1970), 345–358. MR 266241, DOI 10.1016/0040-9383(70)90058-3
Henning Hauschild, Äquivariante Transversalität und äquivariante Bordismentheorien, Arch. Math. (Basel) 26 (1975), no. 5, 536–546. MR 402787, DOI 10.1007/BF01229778
Max Karoubi, $K$-theory, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin-New York, 1978. An introduction. MR 0488029
L. G. Lewis Jr., The uniqueness of bundle transfers, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 1, 87–111. MR 684278, DOI 10.1017/S0305004100060369
L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778
Peter Löffler, Bordismengruppen unitärer Torusmannigfaltigkeiten, Manuscripta Math. 12 (1974), 307–327 (German, with English summary). MR 348776, DOI 10.1007/BF01171078
Goro Nishida, The transfer homomorphism in equivariant generalized cohomology theories, J. Math. Kyoto Univ. 18 (1978), no. 3, 435–451. MR 509493, DOI 10.1215/kjm/1250522505
Christian Okonek, Der Conner-Floyd-Isomorphismus für Abelsche Gruppen, Math. Z. 179 (1982), no. 2, 201–212 (German). MR 645496, DOI 10.1007/BF01214312
Ted Petrie, Pseudoequivalences of $G$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 169–210. MR 520505
Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452
Stefan Waner, Equivariant $R\textrm {O}(G)$-graded bordism theories, Topology Appl. 17 (1984), no. 1, 1–26. MR 730020, DOI 10.1016/0166-8641(84)90019-1
Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150. MR 250324, DOI 10.1016/0040-9383(69)90005-6
Klaus Wirthmüller, Equivariant homology and duality, Manuscripta Math. 11 (1974), 373–390. MR 343260, DOI 10.1007/BF01170239