#### 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 $B\subset N$ be compact smooth manifolds of dimensions $n+k,n$ and $\ell$, respectively. Given a map $f:M\to N$, we give homological conditions under which ${g}^{-1}\left(B\right)$ has nontrivial cohomology (with local coefficients) for any map $g$ homotopic to $f$. We also show that a certain cohomology class in ${H}^{j}\left(N,N-B\right)$ is Poincaré dual (with local coefficients) under ${f}^{\ast }$ to the image of a corresponding class in ${H}_{n+k-j}\left({f}^{-1}\left(B\right)\right)$ 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
Primary: 55M20
Secondary: 55S35