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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A fixed point theorem for manifolds

Author: Jan W. Jaworowski
Journal: Proc. Amer. Math. Soc. 28 (1971), 275-278
MSC: Primary 55.36; Secondary 54.00
MathSciNet review: 0273604
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Abstract: A Lefschetz type fixed point theorem is proved extending a recent theorem by Robert F. Brown. It deals with compact maps of the form $f:(M - U,X) \to (M,M - U)$, where $M$ is an $n$-manifold, $X$ is an $(n - 2)$-connected ANR which is closed in $M$ and $U$ is an unbounded component of $M - U$. The map $f$ defines maps $u:M - U \to M - U$ and $v:M \to M$; the Lefschetz numbers of $u$ and $v$ are defined and are shown to be equal; and if this number is nonzero then $f$ has a fixed point.

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Keywords: Manifold, retraction, absolute neighborhood retract, compact map, Lefschetz map, <IMG WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$\Lambda$">-map, fixed point
Article copyright: © Copyright 1971 American Mathematical Society