Torsion in the bordism of oriented involutions

Rowlett, Russell J.

doi:10.1090/S0002-9947-1977-0445521-2

Torsion in the bordism of oriented involutions

Author: Russell J. Rowlett
Journal: Trans. Amer. Math. Soc. 231 (1977), 541-548
MSC: Primary 57D85
DOI: https://doi.org/10.1090/S0002-9947-1977-0445521-2
Erratum: Trans. Amer. Math. Soc. 248 (1979), 221-221.
MathSciNet review: 0445521
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Abstract: In the bordism theory ${\Omega _ \ast }(Z_2^k)$ of smooth, orientation-preserving -actions all torsion has order two. Furthermore, the torsion classes inject in the unoriented theory ${N_ \ast }(Z_2^k)$ , and any class represented by a stationary-point free action has infinite order. In addition, a procedure is given for producing Smith constructions in some generality.

[1] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
[2] 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
[3] F. Hirzebruch, Involutionen auf Mannigfaltigkeiten, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 148–166 (German). MR 0250339
[4] Richard S. Palais, The classification of 𝐺-spaces, Mem. Amer. Math. Soc. No. 36, 1960. MR 0177401
[5] H. L. Rosenzweig, Bordism of involutions on manifolds, Illinois J. Math. 16 (1972), 1–10. MR 0290386
[6] R. J. Rowlett, Wall manifolds with involution, Trans. Amer. Math. Soc. 169 (1972), 153–162. MR 0314076, https://doi.org/10.1090/S0002-9947-1972-0314076-0
[7] R. J. Rowlett, Bounding a free action of a dihedral group, Michigan Math. J. 21 (1974), 363–370 (1975). MR 0377942
[8] -, Torsion in the bordism of oriented involutions, Notices Amer. Math. Soc. 22 (1975), A227. Abstract #720-57-1.
[9] Robert E. Stong, Notes on cobordism theory, Mathematical notes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0248858
[10] R. E. Stong, Complex and oriented equivariant bordism, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 291–316. MR 0273644
[11] 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
[12] R. E. Stong, Equivariant bordism and (𝑍₂)^{𝑘} actions, Duke Math. J. 37 (1970), 779–785. MR 0271966
[13] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 0120654, https://doi.org/10.2307/1970136

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