On the space of piecewise linear homeomorphisms of a manifold

Abstract

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