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Involutions Fixing Fn ∪ ﹛Indecomposable﹜

Published online by Cambridge University Press:  20 November 2018

Pedro L. Q. Pergher
Pedro L. Q. Pergher*
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, São Carlos, SP 13565- 905, Brazil e-mail: pergher@dm.ufscar.br
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Abstract

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Let ${{M}^{m}}$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\,:\,{{M}^{m}}\,\to \,{{M}^{m}}$ whose fixed point set has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$, where ${{F}^{n}}$ and ${{F}^{j}}$ are submanifolds with dimensions $n$ and $j$, ${{F}^{j}}$ is indecomposable and $n\,>\,j$. Write $n\,-\,j\,={{2}^{p}}q$, where $q\,\ge \,1$ is odd and $p\,\ge \,0$, and set $m(n\,-\,j)\,=\,2n\,+\,p\,-q\,+\,1$ if $p\,\le \,q\,+\,1$ and $m(n\,-\,j)\,=\,2n\,+\,{{2}^{p-q}}$ if $p\,\ge \,q$. In this paper we show that $m\,\le \,m(n-j)+2j+1$. Further, we show that this bound is almost best possible, by exhibiting examples $({{M}^{m(n-j)+2j}},\,T)$ where the fixed point set of $T$ has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$ described above, for every $2\,\le \,j\,<\,n$ and $j$ not of the form ${{2}^{t}}\,-\,1$ (for $j\,=\,0$ and 2, it has been previously shown that $m(n\,-\,j)\,+\,2j$ is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses $m\,\le \,\frac{5}{2}\,n$.

Keywords

57R85involutionprojective space bundleindecomposable manifoldsplitting principleStiefel– Whitney classcharacteristic number
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 164 - 171
Copyright
Copyright © Canadian Mathematical Society 2012

References

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