On the bordism of almost free 𝑍_{2𝑘} actions

Beem, R. Paul

doi:10.1090/S0002-9947-1977-0425991-6

On the bordism of almost free $Z_{2k}$ actions

Author: R. Paul Beem
Journal: Trans. Amer. Math. Soc. 225 (1977), 83-105
MSC: Primary 57D85
DOI: https://doi.org/10.1090/S0002-9947-1977-0425991-6
MathSciNet review: 0425991
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An “almost free” ${Z_{{2^k}}}$ action on a manifold is one in which only the included ${Z_2}$ may possibly fix points of the manifold. For k = 2, these are the stationary-point free actions. It is shown that almost free ${Z_{{2^k}}}$ bordism is generated by three subalgebras: the extension from ${Z_2}$ actions, a coset of ${Z_2}$ extensions being the restrictions of circle actions and a certain ideal of elements which annihilate the whole ring. The additive structure is determined. Free ${Z_{{2^k}}}$ bordism is shown to split as an algebra. It is shown that the kernel of the extension homomorphism from ${Z_2}$ to ${Z_{{2^k}}}$ bordism is equal to the image of the corresponding restriction homomorphism.

R. Paul Beem, The action of free $G$-bordism on $G$-bordism, Duke Math. J. 42 (1975), 297–305. MR 372882

P. E. Conner and E. E. Floyd, Differentiate periodic maps, Ergebnisse Math. Grenzgebiete, N. F., Band 33, Springer-Verlag, Berlin; Academic Press, New York, 1964. MR 31 #750.

Yutaka Katsube, Principal oriented bordism algebra $\Omega _{\ast }(Z_{2}\,k)$, Hiroshima Math. J. 4 (1974), 265–277. MR 348775
C. N. Lee, Equivariant homology theories, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 237–244. MR 0250299
R. J. Rowlett, Bounding a free action of an abelian group, Duke Math. J. 41 (1974), 381–385. MR 341506
R. E. Stong, Equivariant bordism and Smith theory. IV, Trans. Amer. Math. Soc. 215 (1976), 313–321. MR 405464, DOI https://doi.org/10.1090/S0002-9947-1976-0405464-6

---, On the bordism algebra of involutions, Unpublished notes, University of Virginia.

R. E. Stong, Unoriented bordism and actions of finite groups, Memoirs of the American Mathematical Society, No. 103, American Mathematical Society, Providence, R.I., 1970. MR 0273645