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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Wall manifolds with involution


Author: R. J. Rowlett
Journal: Trans. Amer. Math. Soc. 169 (1972), 153-162
MSC: Primary 57D75
MathSciNet review: 0314076
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Abstract: Consider smooth manifolds $ W$ with involution $ t$ and a Wall structure described by a map $ f:W \to {S^1}$ such that $ ft = f$. For such objects we define cobordism theories $ {\text{W}}_\ast ^I$ (in case $ W$ is closed, $ t$ unrestricted), $ {\text{W}}_ \ast ^F$ (for $ W$ closed, $ t$ fixed-point free), and $ {\text{W}}_ \ast ^{{\text{rel}}}$ ($ W$ with boundary, $ t$ free on $ W$). We prove that there is an exact sequence

$\displaystyle 0 \to {\text{W}}_ \ast ^I \to {\text{W}}_ \ast ^{{\text{rel}}} \to {\text{W}}_ \ast ^F \to 0.$

As a corollary, $ {\text{W}}_ \ast ^I$ imbeds in the cobordism of unoriented manifolds with involution. We also describe how $ {\text{W}}_ \ast ^I$ determines the $ 2$-torsion in the cobordism of oriented manifolds with involution.

References [Enhancements On Off] (What's this?)

  • [1] Peter G. Anderson, Cobordism classes of squares of orientable manifolds, Ann. of Math. (2) 83 (1966), 47–53. MR 0187247 (32 #4700)
  • [2] Pierre E. Conner, Lectures on the action of a finite group, Lecture Notes in Mathematics, No. 73, Springer-Verlag, Berlin-New York, 1968. MR 0258023 (41 #2670)
  • [3] P. E. Connor and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 33, Academic Press, New York; Springer-Verlag, Berlin, 1964. MR 31 #750.
  • [4] -, Supplement to differentiable periodic maps, University of Virginia, Charlottesville, Va., 1965. (mimeograph).
  • [5] H. Rosenzweig, Bordism groups of all orientation preserving involutions, Dissertation, University of Virginia, Charlottesville, Va., 1967.
  • [6] Robert E. Stong, Notes on cobordism theory, Mathematical notes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0248858 (40 #2108)
  • [7] 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 (42 #8521)
  • [8] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 0120654 (22 #11403)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0314076-0
PII: S 0002-9947(1972)0314076-0
Keywords: Wall manifold, orientation-preserving involution, equivariant cobordism
Article copyright: © Copyright 1972 American Mathematical Society