Cohomology of preimages with local coefficients

Details Cohomology of preimages with local coefficients Cohomology of preimages with local coefficients

2009-04-27

arxiv.org/abs/0904.4177v1

Algebr. Geom. Topol. 6 (2006) 1471-1489

Mathematics

Geometric Topology

This is the version published by Algebraic & Geometric Topology on 4 October 2006

Scientific paper

10.2140/agt.2006.6.1471

Let M,N and B\subset N be compact smooth manifolds of dimensions n+k,n and \ell, respectively. Given a map f from 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 Poincare 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.

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