Volume 6, issue 3 (2006)

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Cohomology of preimages with local coefficients

Daciberg Lima Gonçalves and Peter Wong

Algebraic & Geometric Topology 6 (2006) 1471–1489

arXiv: 0904.4177

Abstract

Let M,N and B N be compact smooth manifolds of dimensions n + k,n and , respectively. Given a map f : M N, we give homological conditions under which g1(B) has nontrivial cohomology (with local coefficients) for any map g homotopic to f. We also show that a certain cohomology class in Hj(N,NB) is Poincaré dual (with local coefficients) under f to the image of a corresponding class in Hn+kj(f1(B)) when f is transverse to B. This generalizes a similar formula of D Gottlieb in the case of simple coefficients.

Keywords
fibration, local trivial fibration, Poincaré duality, local coefficient system, (co)homology with local coefficients
Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 55S35
References
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Publication
Received: 16 March 2005
Revised: 25 April 2006
Accepted: 14 August 2006
Published: 4 October 2006
Authors
Daciberg Lima Gonçalves
Dept. de Matemática
IME - USP
Caixa Postal 66.281
CEP 05311-970 São Paulo - SP
Brasil
Peter Wong
Department of Mathematics
Bates College
Lewiston, ME 04240
USA