Cohomology of preimages with local coefficients

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.