Cohomology of preimages with local coefficients

Gonçalves, Daciberg Lima; Wong, Peter

doi:10.2140/agt.2006.6.1471

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Cohomology of preimages with local coefficients

Daciberg Lima Gonçalves and Peter Wong

Algebraic & Geometric Topology 6 (2006) 1471–1489

DOI: 10.2140/agt.2006.6.1471

arXiv: 0904.4177

Abstract

Let $M, N$ and $B \subset N$ be compact smooth manifolds of dimensions $n + k, n$ and $ℓ$ , respectively. Given a map $f : M \to N$ , we give homological conditions under which $g^{- 1} (B)$ has nontrivial cohomology (with local coefficients) for any map $g$ homotopic to $f$ . We also show that a certain cohomology class in $H^{j} (N, N - B)$ is Poincaré dual (with local coefficients) under $f^{*}$ to the image of a corresponding class in $H_{n + k - j} (f^{- 1} (B))$ 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

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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

Volume 6, issue 3 (2006)