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Involutions Fixing Fn ∪ ﹛Indecomposable﹜
Published online by Cambridge University Press: 20 November 2018
Abstract
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
$m\,\le \,\frac{5}{2}\,n$.
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- Copyright © Canadian Mathematical Society 2012
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