Let X be a compact PL
manifold and Q denote the Hilbert cube Iω. In this paper, we show that the
following subgroups of the homeomorphism group H(X × Q) of X × Q are
manifolds:
Hfd(X × Q)
= {h ×id|h ∈ H(X × In) for some n ∈ ℕ},
HPL(X × Q)
= {h ×id∈ Hfd(X × Q)|h is PL} and
HLIP(X × Q)
= all Lipschitz homomorphisms of (X × Q)
under some suitably chosen metric.
In fact, let H∗(X × Q) denote the subspace consisting of those homeomorphisms
which are isotopic to a member of H∗(X ×Q), where ∗ =fd, PL or LIP respectively.
Then it is shown that
(HPL(X × Q),HPL(X × Q)) is an (l2,l2f)-manifold pair,
(HLIP(X × Q),HLIP(X × Q)) is an (l2,l2Q)-manifold pair and
Hfd(X × Q) is an (l2× l2f)-manifold and dense in Hfd(X × Q),
where l2 is the separable Hilbert space, l2f= {(xi) ∈ l2|xi= 0 except for finitely many i}
and l2Q= {(xi) ∈ l2|sup|i ⋅ xi| < ∞}.
