Homotopy equivalences of manifolds

Question

Suppose I have a self homotopy equivalence f of an oriented manifold $M^n$ such that f is the identity map on homology of degree $\le n-2$. When is this map necessarily the identity map on all of homology? This is not true if f reverses orientation- for example, the antipodal map on even dimensional spheres satisfied the conditions but is not an isomorphism on nth degree homology. If f is an orientation preserving diffeomorphism, then the claim is also true since the mapping torus is an oriented manifold; however I don't know the answer for general homotopy equivalences.

Answer 1

Assuming $M$ is closed and connected and $n>2$, any such orientation-preserving $f$ is the identity on homology. This is basically immediate from Poincare duality: cap product with the fundamental class induces an isomorphism $H^1(M)\to H_{n-1}(M)$, and $f$ preserves the fundamental class since it is orientation-preserving. Since $n>2$, $f$ induces the identity on $H_1$ and hence also on $H^1$. It follows by naturality of the cap product that $f$ induces the identity on $H_{n-1}$.

For $n=2$, this is not true, even for diffeomorphisms. For instance, if $M$ is the torus $\mathbb{R}^2/\mathbb{Z}^2$, any element of $SL_2(\mathbb{Z})$ induces an orientation-preserving diffeomorphism of $M$, and its induced map on $H_1(M)=\mathbb{Z}^2$ is the usual map induced by the matrix.