Bundle transfer of \(L\)-homology orientation classes for singular spaces (Q6614581)

When it exists, the transfer map \(p^!\) of a morphism \(p\) (whatever category contains \(p\)) plays an essential role, see \textit{J. C. Becker} and \textit{D. H. Gottlieb}, in: History of topology. Amsterdam: Elsevier. 725--745 (1999; Zbl 0957.55001)]. For instance, let us consider a fiber bundle, \(p\colon X\to B\), whose structure group is a compact Lie group acting smoothly on a compact \(d\)-dimensional manifold fiber \(F\) and whose base space is a finite complex. When \(B\) is a closed oriented manifold, under usual orientation conditions, it is known that the transfer map \(p^!\colon H_{n}(B)\to H_{n+d}(X)\) sends the fundamental class of \(B\) to the fundamental class of \(X\). As a corollary, with Whitney sum formula, the cohomological Hirzebruch \(L\)-class, \(L^*(T_{p})\), of the vertical bundle of \(p\), and the Poincaré duals, \(L_{*}(-)\in H_{*}(-;\mathbb Q)\), of the cohomological Hirzebruch \(L\)-classes of the tangent bundles verify \[p^! L_{*}(B)= L^*(T_{p})^{-1}\cap L_{*}(X).\]\N\NThe main objective of the work under review is the achievement of similar equality within the framework of pseudo-manifolds. M. Goresky and R. MacPherson define \(L\)-cohomology classes when the pseudo-manifold \(B\) has only even codimensional strata in [\textit{M. Goresky} and \textit{R. MacPherson}, Topology 19, 135--165 (1980; Zbl 0448.55004)]. This has been extended to more general situation later, as in [\textit{M. Banagl}, Ann. Math. (2) 163, No. 3, 743--766 (2006; Zbl 1101.57013)]. Unfortunately, the argument mentioned at the beginning does not apply here. The author is therefore required to construct a transfer map, \(\xi^!\colon E_{n}(B)\to E_{n+d}(X)\), for any module spectrum \(E\) over the Thom spectrum. This induces transfer map on Witt bordism theory with the same behavior as \(p^!\) with respect to fundamental classes. Finally, the equality sought with \(L\)-classes is obtained from [\textit{M. Banagl} et al., Sel. Math., New Ser. 25, No. 1, Paper No. 7, 104 p. (2019; Zbl 1416.55001)] and an explicit writing for the transfer after tensorization with the rationals.