Homology, Homotopy and Applications

Umkehr maps

Ralph L. Cohen and John R. Klein

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Abstract

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.

Article information

Source
Homology Homotopy Appl., Volume 11, Number 1 (2009), 17-33.

Dates
First available in Project Euclid: 1 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.hha/1251832558

Mathematical Reviews number (MathSciNet)
MR2475820

Zentralblatt MATH identifier
1160.55005

Subjects
Primary: 55N99: None of the above, but in this section
Secondary: 55M05: Duality 55R70: Fibrewise topology

Keywords
Umkehr map fibered spectrum Poincaré duality string topology

Citation

Cohen, Ralph L.; Klein, John R. Umkehr maps. Homology Homotopy Appl. 11 (2009), no. 1, 17--33. https://projecteuclid.org/euclid.hha/1251832558


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