The space of simplexwise linear homeomorphisms of a convex 2-disk

Strictly convex simplexwise linear embeddings of a $2$-disk

“…(A map from a triangulated 2-disk into R2 is called simplexwise linear (SL) if it is affine linear on each (closed) 2-simplex.) Our results are the analogs in the strictly convex case of [BCH,Theorem 5.1] and [B,Theorem 1.2]. The interest in strictly convex SL embeddings is threefold.…”

“…First, for a fixed x G S1, one can deduce the existence of local collarings using the convex disk decomposition method of [BCH,§4]. The existence of the global collaring follows from the local one by standard arguments (as in [Br]).…”

Spaces of geodesic triangulations of the sphere

“…Thus f >-* p°f defines a homeomorphism of GT(ff; S, c) onto the space E of strictly convex triangulated 2-cells in R2 which are orientation-preserving isomorphic to K \ st(a), contain the origin, and fix the vertex ai G st (o) on a given ray. This space, E is easily seen to be homotopy equivalent to the space D defined in the proof of Theorem 1.1 in §4 [Bloch, 1985]. Ethan Bloch shows that D (and, therefore, E and GT(Ä"; S, c)) are contractible.…”

“…In §3 we restrict our attention to n -2 and use the results of §2 to show that G(Long) = {g G GT(K; ct) | for some longitude I, each 1-simplex in g(K) intersecting / either has a vertex on / or is contained in /} deforms into G(Long) n GT(K; S, e) where GT{K; S, c) = {g G GT(K; S) | ff(link(a)) is strictly convex}. A theorem of Ethan Bloch's [Bloch, 1985] concerning strictly convex embeddings of a triangulated 2-cell into R2 implies that (for n = 2) GT(K; S, c) is contractible. As a consequence of Bloch's result and of 2.5 and 3.3 below, we have THEOREM, (a) G(Long) is contractible to a point in G(Long) U GT(K; S,c) C GT(K;a).…”

“…Spaces of geodesic triangulations of spheres are related to smoothings of combinatorial manifolds in [Cairns, 1940], [Whitehead, 1961], and [Kuiper, 1965].…”