On the transfer map of Becker and Gottlieb
In [J. Differ. Geom. 10, 39--48 (1975; Zbl 0294.55009)] \textit{D. H. Gottlieb} defines a transfer map \(\tau :H^{*}(E,\mathbb{Z})\rightarrow H^{*}(B,\mathbb Z)\) for a fibration \(F@>i >>E@>p >>B\), where the fibre \(F\) is a compact, connected and orientable manifold. This transfer map generalizes a covering map. In [Compos. Math. 33, 107--133 (1976; Zbl 0354.55009)] \textit{D. H. Gottlieb} and \textit{J. C. Becker} extend the transfer map to any fibration which has as a fiber a finite CW-complex. A survey on this subject is [\textit{J. M. Lemaire}, Sémin. Bourbaki 1975/76, Lect. Notes Math. 567, Exp. No. 472 (1977; Zbl 0344.55017)]. In the paper under review, a fibration \(F@>i >>E@>p >>B\) is considered, such that \(E\) and \(B\) are in the Anick range (i.e. they are \(1\)-connected and their homologies \(H^{*}(E,\mathbf{k})\) and \( H^{*}(B,\mathbf{k})\) respectively are concentrated in the degrees from \(r\) to \(r\rho (\mathbf{k})\) for an \(r\geq 1\), where \(\rho (\mathbf{k})\) is the characteristic of the fixed field \(\mathbf{k}\)) and \(H^{*}(F,\mathbf{k})\) has a finite dimension. Using a KS-extension (or a relative Sullivan model) of the projection \(p\), one shows that the transfer map exists at the cochain level. Using the same ideas, a result concerning the inclusion of the fiber is given, when the cohomology of the fiber satisfies Poincaré duality.
scientific article; zbMATH DE number 1421348