Fixed point problems, equivariant stable homotopy, and a trace map for the algebraic \(K\)-theory of a point
Let \(A(*)\) be the Waldhausen \(A\)-functor applied to the one-point space. The cyclic group \(\mathbb{Z}/k\) acts on \(\mathbb{R}^{nk} = \prod^k_{i=1} \mathbb{R}^n\) by permuting the \(k\) factors. This induces a \(\mathbb{Z}/k\)-action on the one-point compactification \(S^{nk} = \mathbb{R}^{nk} \cup \{\infty\}\) which fixes the base point \(\infty\) in \(S^{nk}\). We obtain an action of \(\mathbb{Z}/k\) on \(\Omega^{nk}Z\) which is the space of pointed maps from \(S^{nk}\) to \(Z\). Taking the colimit over the \(\mathbb{Z}/k\)-spaces \(\Omega^{nk} S^{nk}\) yields a \(\mathbb{Z}/k\)-space denoted by \(Q^{nk} (*)\). If we forget the \(\mathbb{Z}/k\)-action, \(Q^{\mathbb{Z}/k} (*)\) is just a model for \(Q(*)\). In the paper an equivariant trace map \(\text{tr} : A(*) \to Q^{\mathbb{Z}/k} (*)\) is constructed. In particular one obtains for \(k=1\) a map \(\pi_0 (A(*)) \to \pi_0(Q(*))\) which is compatible with the isomorphisms from \(\pi_0 (A(*))\), resp. \(\pi_0(Q(*))\), to \(\mathbb{Z}\) which assigns to a pointed CW-complex \((X,x)\) representing an element in \(\pi_0(A(*))\) its Euler characteristic \(\chi(X,x)\) resp. which assigns to a selfmap \(g\) of \(S^n\) representing an element in \(\pi_0 (Q(*))\) its degree. The passage from a selfmap \(f:X \to X\) of a finite CW-complex to a selfmap \(g:S^n \to S^n\) is roughly as follows. Embed \(X\) into \(\mathbb{R}^n\) and choose a neighborhood \(U\) with retraction \(r:U \to X\). Then let \(g\) be any pointed map \(S^n \to S^n\) which takes the value zero on the fixed point set of \(f\) and looks near the fixed point set like \(g(x)=x-f\circ r(x)\). Then the pointed homotopy type of \(g\) is already determined by these requirements. The construction of the more general trace on space level taking the \(\mathbb{Z}/k\)-action into account is based on an extension of the passage from \(X\) to \(g\) mentioned above to circular diagrams \(X_1 @>f_1>> X_2 @>f_2>> \cdots @>f_{k-1}>> X_k @>f_k>> X_1\) and considering the special case where all \(f_i\) are the same.
scientific article; zbMATH DE number 863711