This is a question from Massey's Singular Homology Theory. It reads:
Let $M_1$ and $M_2$ be closed orientable $n$-manifolds, and let $f : M_1 \to M_2$ be a continuous map such that $f_* : H_n(M_1) \to H_n(M_2)$ is an isomorphism.
Prove that the induced homomorphism $f_* : H_k(M_1) \to H_k(M_2)$ is an epimorphism and the kernel of $f_*$ is a direct summand of $H_k(M_1)$.
Similarly, prove that $f^*: H^k(M_2) \to H^k(M_1)$ is a monomorphism and the image of $f^*$ is a direct summand of $H^k(M_1)$.
I already showed that $f_*$ is an epimorphism and $f^*$ is a monomorphism. However, I don't know how to prove that $\operatorname{ker} f_*$ and $\operatorname{im} f^*$ are direct summands of $H_k(M_1)$ and $H^k(M_1)$, respectively. Could you help?
This follows from Poincaré duality and the projection formula for the cap product. Pick orientations on $M_1$ and $M_2$ such that $f$ maps the fundamental class $[M_1]$ to the fundamental class $[M_2]$. Let $D_i:H^k(M_i)\to H_{n-k}(M_i)$ be the Poincaré duality isomorphism, given explicitly by $D_i(a)=[M_i]\cap a$. By the projection formula, for $a\in H^k(M_2)$ we then have $$D_2(a)= [M_2]\cap a=f_*([M_1])\cap a=f_*([M_1]\cap f^*(a))=f_*(D_1(f^*(a))).$$
That is, $D_2=f_*\circ D_1\circ f^*$, so $1=D_2\circ D_2^{-1}=f_*\circ D_1\circ f^*\circ D_2^{-1}$. This shows that $f_*$ has a right inverse $D_1\circ f^*\circ D_2^{-1}$, so $f_*$ is a split surjection. Similarly, $1=D_2^{-1}\circ D_2=D_2^{-1}\circ f_*\circ D_1\circ f^*$ so $f^*$ has a left inverse and is a split injection.