Fixed point sets of equivariant fiber-preserving maps.

*(English)*Zbl 1390.55004Given a selfmap \(f: X\to X\) on a compact connected polyhedron \(X\), H. Schirmer in [Topology Appl. 37, No. 2, 153–162 (1990; Zbl 0717.55001)] gave necessary and sufficient conditions for a nonempty closed subset \(A\) to be the fixed point set of a map in the homotopy class of \(f\). The result was extended by R. F. Brown and C. L. Soderlund in [J. Fixed Point Theory Appl. 2, No. 1, 41–53 (2007; Zbl 1128.55001)] to the category of fiber bundles and fiber-preserving maps. The goal of the work under review is to prove equivariant analogues of the results from the above-referenced papers. One result is about \(G\)-equivariant maps for \(G\) a compact Lie group. The second is about \(G\)-fiber maps on \(G\)-fiber bundles for \(G\) a finite group. The authors assume further hypotheses coming from the known results, namely the spaces are assumed to be compact and smooth \(G\)-manifolds. In the first case they show:

Theorem 1.5. Let \(G\) be a compact Lie group, \(X\) be a compact and smooth \(G\)-manifold and \(A\) be a nonempty, closed, locally contractible \(G\)-subset of \(X\) such that for each finite \(WK\) we assume that \(\dim(X^K) \geq 3\), \(\dim(X^K) - \dim(X^K - X_K) \geq 2\) and \(A^K\) is by-passed in \(X^K\), for all \((K) \in \mathrm{Iso}(X)\). Suppose that the following conditions holds for a \(G\)-map \(f: X \to X\):

Using Theorem 1.5 the authors show the second main result, Theorem 1.6, which is about the fixed point set of fibre maps and the group \(G\) is assumed to be finite. The statement of the result involves too much technical notation and we do not state it here. The proofs are not straightforward and use non-trivial machinery of \(G\)-equivariant maps in the smooth category. The paper is well organized and details are provided.

Theorem 1.5. Let \(G\) be a compact Lie group, \(X\) be a compact and smooth \(G\)-manifold and \(A\) be a nonempty, closed, locally contractible \(G\)-subset of \(X\) such that for each finite \(WK\) we assume that \(\dim(X^K) \geq 3\), \(\dim(X^K) - \dim(X^K - X_K) \geq 2\) and \(A^K\) is by-passed in \(X^K\), for all \((K) \in \mathrm{Iso}(X)\). Suppose that the following conditions holds for a \(G\)-map \(f: X \to X\):

- (CG1)
- there exists a \(G\)-homotopy \(H_A: A \times I \to X\) from \(f|_A\) to the inclusion \(i: A \hookrightarrow X\);
- (CG2)
- for each finite \(WK\), for every \(WK\)-essential fixed point class \(F\) of \(f^K : X^K \to X^K\) there exists a path \(\alpha : I \to X^K\) with \(\alpha(0) \in F\), \(\alpha(1) \in A^K\), and \(\{\alpha(t)\} \sim \{ f^K \circ ( t ) \} * \{ H_A^K ( \alpha(1) , t ) \}\).

Using Theorem 1.5 the authors show the second main result, Theorem 1.6, which is about the fixed point set of fibre maps and the group \(G\) is assumed to be finite. The statement of the result involves too much technical notation and we do not state it here. The proofs are not straightforward and use non-trivial machinery of \(G\)-equivariant maps in the smooth category. The paper is well organized and details are provided.

Reviewer: Daciberg Lima Gonçalves (São Paulo)