On the space of piecewise linear homeomorphisms of a manifold

Geoghegan, Ross; Haver, William E.

doi:10.1090/S0002-9939-1976-0402785-3

On the space of piecewise linear homeomorphisms of a manifold
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by Ross Geoghegan and William E. Haver

Proc. Amer. Math. Soc. 55 (1976), 145-151

DOI: https://doi.org/10.1090/S0002-9939-1976-0402785-3

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Abstract:

Let ${M^n}$ be a compact PL manifold, $n \ne 4$; if $n = 5$, suppose $\partial M$ is empty. Let $H(M)$ be the space of homeomorphisms on $M$ and ${H^{\ast }}(M)$ the elements of $H(M)$ which are isotopic to PL homeomorphisms. It is shown that the space of PL homeomorphisms, $PLH(M)$, has the finite dimensional compact absorption property in ${H^{\ast }}(M)$ and hence that $({H^{\ast }}(M),PLH(M))$ is an $({l_2},l_2^f)$-manifold pair if and only if $H(M)$ is an ${l_2}$-manifold. In particular, if ${M^2}$ is a $2$-manifold, $(H({M^2}),PLH({M^2}))$ is an $({l_2},l_2^f)$-manifold pair.

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