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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 \[ 0 \to {\text {W}}_ \ast ^I \to {\text {W}}_ \ast ^{{\text {rel}}} \to {\text {W}}_ \ast ^F \to 0.\] 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.

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  • Peter G. Anderson, Cobordism classes of squares of orientable manifolds, Ann. of Math. (2) 83 (1966), 47–53. MR 187247, DOI
  • 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
  • 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. ---, Supplement to differentiable periodic maps, University of Virginia, Charlottesville, Va., 1965. (mimeograph). H. Rosenzweig, Bordism groups of all orientation preserving involutions, Dissertation, University of Virginia, Charlottesville, Va., 1967.
  • 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, 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
  • C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 120654, DOI

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Keywords: Wall manifold, orientation-preserving involution, equivariant cobordism
Article copyright: © Copyright 1972 American Mathematical Society