On the space of piecewise linear homeomorphisms of a manifold

Abstract

Let Af" be a compact PL manifold, n # 4; if n = 5, suppose 3M is empty. Let H(M) be the space of homeomorphisms on M and H* (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*(M) and hence that (H*(M),PLH(M)) is an (/2,//)-manifold pair if and only if H(M) is an l2manifold. In particular, if M* is a 2-manifold, (H(M2),PLH(M2)) is an (/2,//)-manifold pair. PLHiM) X //is an //-manifold. Finally, Keesling and Wilson [16] have shown that PLHiM) X l{ is homeomorphic to PLHiM). Hence PLHiM) is an l{manifold. In general it is not true that PLHiM) is dense in HiM). For example, Kirby and Siebenmann (see [17]) have shown that HiS2 X S3) has a component containing no PL homeomorphism. They have also shown that if the

Diffeomorphism groups of non-compact manifolds endowed with the WhitneyC∞-topology

“…Theorem 6.6. ( [15], [23], cf. [32]) Suppose M is a compact 2-manifold possibly with boundary and K M is a subpolyhedron.…”

On Hyperspaces of Polyhedra

“…Note that in Lemma 3.1 A% c Pm if sé c Pm . Similar to [GH,Lemma 2], by using Lemmas 3.4 and 3.1, we can construct a map g : A -> Pm of A into some Pm D Pn such that pH(g, f) < e/2 and g\B = f\B . As in the proof of [CN,Lemma 4.6], by using Lemma 3.3 we can replace g by an embedding h : A -► P. of A into some P¡ D Pm such that h\B = g\B = f\B and pH(h , g) < e/2 so pH(h , f) < e .…”

Topological (\prod^ω\ell_2, \sum^ω\ell_2)-factors of diffeomorphism groups of non-compact manifolds

“…In [20] we obtained a general characterization of infinite-dimensional manifold tuples based upon the stability property (cf. [7,15,19], [1,4,5,6], [11], etc.). In this section we deduce a characterization of ( ω ℓ 2 , ω ℓ 2 )-manifolds from this general characterization theorem.…”