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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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