Umkehr maps. (English) Zbl 1160.55005

The authors give a unified treatment of umkehr homomorphisms such as those defined by the Pontryagin-Thom construction, by intersections of chains, by integration along fibres, by push-forward constructions, and by the Becker-Gottlieb transfer. They define an umkehr functor to be a contravariant functor from manifolds to spectra satisfying certain axioms, they show that these are simply the functors of the form \(\text{map}(-,E)\), and they show how various umkehr homomorphisms arise from various choices of \(E\). Because of applications in string topology, they also give the theory in a fibrewise setting, proving a fibrewise version of Brown representability.


55N99 Homology and cohomology theories in algebraic topology
55M05 Duality in algebraic topology
55R70 Fibrewise topology
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