Cohomology of preimages with local coefficients

TL;DR

This paper investigates the conditions under which the preimages of a submanifold under a map have nontrivial cohomology with local coefficients, generalizing classical formulas to more complex coefficient systems.

Contribution

It extends the cohomological analysis of preimages to local coefficients and establishes a Poincaré duality relation in this context.

Findings

Provides homological conditions for nontrivial cohomology of preimages.

Shows a Poincaré duality with local coefficients for transversely intersecting maps.

Generalizes Gottlieb's formula to local coefficient systems.

Abstract

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.