Simplexwise linear near-embeddings of a 2-disk into \({\mathbb{R}}^ 2\). II (Q1088206)

[For part I, see Trans. Am. Math. Soc. 288, 701-722 (1985; Zbl 0563.57008).] Let \(K\subset {\mathbb{R}}^ 2\) be a finitely triangulated 2-disk; a map \(f: K\to {\mathbb{R}}^ 2\) is called simplexwise linear (SL) if \(f| \sigma\) is affine linear for each (closed) 2-simplex \(\sigma\) of K. Let \(L(K)=\{SL\) homeomorphisms \(K\to K\) fixing \(\partial K\) pointwise\(\}\), and let \(\overline{L(K)}\) denote its closure in the space of all SL maps \(K\to {\mathbb{R}}^ 2\). Previous results show that if K is convex, then L(K) is homeomorphic to an open ball in \({\mathbb{R}}^{2k}\), where k is the number of interior vertices of K. The proof of that result made essential use of \(\overline{L(K)}\), causing the topological nature of \(\overline{L(K)}\) to be of interest. For arbitrary K, \(\overline{L(K)}\) need not even be connected. However, a characterization of elements of \(\overline{L(K)}\) is used to show that if K is strictly convex, then \(\overline{L(K)}\) is simply connected.