Algebraic & Geometric Topology 6 (2006) 1471–1489

DOI: 10.2140/agt.2006.6.1471

arXiv: 0904.4177

Abstract

Let M,N and B⊂ N be compact smooth manifolds of dimensions n+k,n and l, respectively. Given a map f:M → 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

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