differential topology - a small questions about hopf theorem

I realize that this doesn't answer your question, but there is also an approach using the methods of homotopy theory and CW complexes. If $M$ is a closed smooth orientable $p$-manifold, then $M$ is homeomorphic to a finite CW complex with cells of dimension $leq p$, and $H^p(M)=mathbb{Z}$.

We may construct a $K(mathbb{Z},p)$ as a CW complex by taking $S^p$ and attaching cells of dimension $geq p+2$ to kill the higher homotopy groups (for example, $K(mathbb{Z},2)= mathbb{CP}^{infty}$ has a cell decomposition with one cell in every even dimension, and $2$-skeleton $S^2$). Let $i:S^phookrightarrow K(mathbb{Z},p)$ be the induced inclusion, representing the fundamental class $[S^p]in H^p(S^p)$.

If we have maps $f_i: M to S^p$, $i=0,1$, such that the induced maps $H^p(f_i): H^p(S^p)to H^p(M)$ are equal, then this implies that $H^p(f_0)([S^p])=H^p(f_1)([S^p])$, and therefore that the maps $icirc f_i: Mto K(mathbb{Z},p)$ are homotopic (by Brown Representability, Thm. 4E.1 Hatcher), and thus are realized by a map $F: Mtimes [0,1] to K(mathbb{Z},p)$, $F_{| Mtimes i}=f_i$. By the Cellular Approximation Theorem (Theorem 4.8 Hatcher), the map $F$ may be homotoped rel $Mtimes { 0, 1 }$ to a map $F': Mtimes [0,1] to K(mathbb{Z},p)^{(p+1)}=S^p$, the $p+1$ skeleton of $K(mathbb{Z},p)$, and thus $f_0simeq f_1$.