Homology, Homotopy and Applications

Volume 11 (2009)

Number 1

Umkehr maps

Pages: 17 – 33

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n1.a2


Ralph L. Cohen (Department of Mathematics, Stanford University, Stanford, Calif., U.S.A.)

John R. Klein (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)


In this note, we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting.


umkehr map, fibered spectrum, Poincaré duality, string topology

2010 Mathematics Subject Classification

Primary 55Nxx. Secondary 55M05, 55R70.

Published 1 January 2009