Generalized transfer and spectral sequences
The author shows that the previously defined generalized fixed-point transfer of \textit{C. Prieto} and \textit{H. Ulrich} [Trans. Am. Math. Soc. 328, No. 2, 731-745 (1991; Zbl 0738.54020)] behaves well with respect to spectral sequences for generalized homology and cohomology theories. Given a metric space \(X\), a \(k\)-fixed point situation over \(X\) is a map \(f:V\to \mathbb{R}^{n +k} \times E\) over \(X\), with \(k\in\mathbb{Z}\) and \(V\) an open subset of \(\mathbb{R}^n \times E\), where \(f\) respects the projection map \(p\circ\text{proj}_E\) with \(p: E\to X\) an ENR over \(X\), and \(\text{Fix}(f)= \{(y,e)\in Y:f(y,e)= (0,e)\}\) lies properly over \(X\), i.e. the the preimage of every compact set in \(X\) is compact in \(\text{Fix} (f)\). The generalized transfer associated to this situation is a homomorphism \(\breve h^*(\text{Fix} (f))\to h^{*+k}(X)\), where \(h^*\) is any generalized cohomology theory and \(\breve h^*\) is its Čechification. Analogously, such a transfer can also be defined for a generalized homology theory \(h_*\). The main theorem of this well written and interesting paper under review is that this general transfer is compatible with the Leray-Serre spectral sequences of a fibration, the Atiyah-Hirzebruch-Whitehead spectral sequences and the Rothenberg-Steenrod spectral sequence associated to the Milnor \(G\)-resolution for the classifying bundle of a topological group. This generalizes analogous results on the compatibility of the Dold and Becker-Gottlieb transfer previously obtained by the author [ibid. 271, 133-142 (1982; Zbl 0491.55009): Bol. Soc. Mat. Mex., II. Ser. 30, No. 2, 37-41 (1985; Zbl 0643.55012)].
scientific article; zbMATH DE number 1787514