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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A fixed point theorem for $ (n-2)$-connected $ n$-polyhedra


Author: Roger Waggoner
Journal: Proc. Amer. Math. Soc. 33 (1972), 143-145
MSC: Primary 55C20
MathSciNet review: 0293622
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Abstract: The main result of this paper is that, for $ n \geqq 4$, a finite $ (n - 2)$-connected polyhedron K of dimension n admits a fixed point free map if either $ {H_{n - 1}}(K;Q)$ or $ {H_n}(K;Q)$ is nonzero, where Q is the field of rational numbers. This result is obtained by first retracting K onto a subpolyhedron C of dimension n or $ n - 1$ which has no local separating points. It is then shown that C admits a map with Lefschetz number zero, and it follows from a theorem of Shi that C does not have the fixed point property. The proof involved may also be applied when K is a 3-dimensional simply connected polyhedron and the subpolyhedron C is also of dimension 3.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0293622-5
PII: S 0002-9939(1972)0293622-5
Keywords: Local separating point, mapping cylinder, retraction, fixed point property
Article copyright: © Copyright 1972 American Mathematical Society