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

Waggoner, Roger

doi:10.1090/S0002-9939-1972-0293622-5

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

Proc. Amer. Math. Soc. 33 (1972), 143-145

DOI: https://doi.org/10.1090/S0002-9939-1972-0293622-5

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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.

References

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