Oriented and weakly complex bordism algebra of free periodic maps

Shibata, Katsuyuki

doi:10.1090/S0002-9947-1973-0315734-5

Oriented and weakly complex bordism algebra of free periodic maps
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by Katsuyuki Shibata PDF

Trans. Amer. Math. Soc. 177 (1973), 199-220 Request permission

Abstract:

Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly complex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps.

References

M. F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200–208. MR 126856, DOI 10.1017/s0305004100035064
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
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
P. E. Conner and E. E. Floyd, The relation of cobordism to $K$-theories, Lecture Notes in Mathematics, No. 28, Springer-Verlag, Berlin-New York, 1966. MR 0216511, DOI 10.1007/BFb0071091
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
Charles H. Giffen, Weakly complex involutions and cobordism of projective spaces, Ann. of Math. (2) 90 (1969), 418–432. MR 253346, DOI 10.2307/1970744
Masayoshi Kamata, The structure of the bordism group $U_\ast (BZ_{p})$, Osaka Math. J. 7 (1970), 409–416. MR 275447
Tsunekazu Kambe, The structure of $K_{\Lambda }$-rings of the lens space and their applications, J. Math. Soc. Japan 18 (1966), 135–146. MR 198491, DOI 10.2969/jmsj/01820135
Katsuhiro Komiya, Oriented bordism and involutions, Osaka Math. J. 9 (1972), 165–181. MR 307221
S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855–951 (Russian). MR 0221509
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
R. E. Stong, Bordism and involutions, Ann. of Math. (2) 90 (1969), 47–74. MR 242170, DOI 10.2307/1970681
J. C. Su, A note on the bordism algebra of involutions, Michigan Math. J. 12 (1965), 25–31. MR 179798
Fuichi Uchida, Bordism algebra of involutions, Proc. Japan Acad. 46 (1970), 615–619. MR 287555
James W. Vick, An application of $K$-theory to equivariant maps, Bull. Amer. Math. Soc. 75 (1969), 1017–1019. MR 245023, DOI 10.1090/S0002-9904-1969-12344-X
C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 120654, DOI 10.2307/1970136
Ching-Mu Wu, Bordism and maps of odd prime period, Osaka Math. J. 8 (1971), 405–424. MR 310862
Tomoyoshi Yoshida, On the $K$-theoretic characteristic numbers of weakly almost complex manifolds with involution, J. Math. Soc. Japan 24 (1972), 527–538. MR 309126, DOI 10.2969/jmsj/02440527
G. G. Kasparov, Invariants of classical lens manifolds in cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 735–747 (Russian). MR 0275449