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Cohomology of preimages with local coefficients. (English) Zbl 1129.55001
Summary: 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.

##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 55S35 Obstruction theory in algebraic topology
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