A polyhedral transversality theorem for one-parameter fixed point theory (Q1365137)

The fixed point set, \(\text{Fix} (h)\), of a piecwise linear map \(h:P \times I\to P\) is the set of points where \(h\) coincides with the projection \(\pi: P\times I\to P\). If \(K\) is a finite simplicial complex, \(K_0\) a simplicial subdivision of \(K\times I\), \(K_0'\) a subdivision of a barycentric subdivision of \(K_0\), and \(f:K_0'\to K\) a simplicial map, and if \(|\pi |: |K_0' |\to |K|\) denotes the projection, then \(\text{Fix} (|f|) =\{x\in |K_0 |: |f|(x) =|\pi|(x)\}\). The author proves that \(\dim (\text{Fix} (|f|)) \leq 1\). Moreover, if \(N\subset |K_0' |\) is the union of the stars of all open simplexes, which meet \(\text{Fix} (|f|)\), and \({\mathcal U}\) is the cover of \(|K|\) by the closures of the stars of vertices of \(K\), then there is a subdivision \(K_0'\) of \(K_0'\) and a simplicial map \(g:K_0' \to K\), with \(|g|\) \({\mathcal U}\)-homotopic to \(|f |\) by the homotopy \(\text{rel cl} (|K_0' |\smallsetminus N)\), such that \(\text{Fix} (|g|) \subset S_0 \cup S_1\), where \(S_1= |\pi |^{-1} (\bigcup\{s \in K:lk(s,K)= \emptyset\})\), \(S_0=|\pi|^{-1} (\bigcup \{s\in K:lk(s,K) \neq\emptyset\) but not connected\}), and \(\dim(S_1\cap \text{Fix} (|g|)) =0\) or 1.