A generalization of the Kuiper-Massey theorem (Q1896972)

Let \(k\) be the real, complex, or quaternion field and \(V\) an \(n\)-dimensional vector space over \(k\). A simplex is said to be coherent if the subspaces in \(V\) associated with its vertices form a complete flag. Let \(\Theta(V)\) denote the union of all coherent simplices in the join \(G_1(V) * G_2(V) *\dots* G_{n-1}(V)\). \textit{V. A. Vassil'ev} [St. Petersbg. Math. J. 3, No. 4, 809-515 (1992); translation from Algebra Anal. 3, No. 4, 113-120 (1991; Zbl 0746.57019)]\ proved that the space \(\Theta(V)\) is homeomorphic to the sphere \(S^M\), where \(M = \dim_\mathbb{R}(k) {n \choose 2} + n - 2\). In this article we construct a homeomorphism \(\Theta(V) \to S^N\) semialgebraic on all strata of \(\Theta(V)\). According to the Tarski-Seidenberg theorem, we obtain the following corollary: Theorem. \(\Theta(V) \overset {\text{PL}} \simeq S^M\).