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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

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

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?)

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