Connectivity properties for subspaces of function spaces determined by fixed points
A question of Schirmer, answered in the negative by the second author [Topology Appl. 37, 293-297 (1990; Zbl 0715.55003)], was, whether, given two homotopic fixed point free selfmaps of a space, there always exists a homotopy connecting them through fixed point free maps. Let \(C(X)\) denote the space of selfmaps of a space \(X\) with the compact-open topology. The authors define \(C_{\min}(X,f)\) for \(f\in C(X)\) to be the subspace of \(C(X)\) of maps that are homotopic to \(f\) and have the minimum number of fixed points among all maps homotopic to \(f\). In the case of \(f\) a fixed point free map, \(X_{\min}(X,f)\) would consist of all the fixed point free maps homotopic to \(f\) and Schirmer's question asks whether that subspace is path connected. A related question is whether \(C_{\min}(X,f)\) is locally path connected. That is, given two sufficiently close maps in \(C_{\min}(X,f)\), is there a small homotopy through maps with minimal fixed point sets between them. Results in an earlier paper of these authors [Fundam. Math. 172, 99-106 (2002; Zbl 0988.55002)] imply that, for \(T\) the 2-torus, the space \(C_{\min}(T,F)\) is path connected and locally path connected if and only if the Lefschetz number of the map \(f\) is non-zero. In the present paper, the authors consider cases where \(X\) is a disc \(D^k\) or sphere \(S^k\). They prove that \(C_{\min}(D^k,f)\) is path connected and locally path connected for all \(k\). They use the \(H\)-space structure on \(S^k\) for \(k=1,3\) and 7 to prove that, for these dimensions, \(C_{\min}(S^k,f)\) is path connected and locally path connected for all \(f\in C(S^k)\).
scientific article; zbMATH DE number 1883742