One-parameter fixed point indices
Let \(F: X \times I \to X\) be a PL homotopy, where \(X\) is a compact connected PL \(n\)-dimensional manifold, in the Euclidean space \(\mathbb{R}^ n\), \(n \geq 4\), and let \(P: X \times I \to X\) be the projection. A fixed point of \(F\) is a point \((x,t)\in X \times I\) such that \(F(x,t) = x\). The set of all the fixed points of \(F\) is denoted by \(\text{Fix}(F)\). For a family \(V\) of isolated circles of fixed points of \(F\) we define two indices: \(\text{ind}_ 1(F,V)\) -- which is an element in the first homology group \(H_ 1(E)\), where \(E\) is the space of paths in \(X \times I \times X\) from the graph of \(F\) to the graph of \(P\); and \(\text{ind}_ 2(F,V)\) -- which is an element in the group \(\mathbb{Z}_ 2\) with two elements. We prove that there is a compact neighborhood \(N\) of \(V\) and a homotopy from \(F\) to \(H\) rel \(X \times I \setminus N\) such that \(\text{Fix}(H) = \text{Fix}(F) \setminus V\) if and only if \(\text{ind}_ 1(V,F) = 0\) and \(\text{ind}_ 2(V,F) = 0\). The indices \(\text{ind}_ 1(F,V)\) and \(i_ 2(V,F)\) are defined via the degrees, \(\deg_ 1(g)\) and \(\deg_ 2(g)\), for maps \(g: S^ 1 \times S^ m \to S^ m\). Moreover, we show how to modify \(F\) to create circles of fixed points with prescribed indices.
scientific article; zbMATH DE number 597578