See here for related.
Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose $M$ is oriented with fundamental class $z$. Let $f: S^n \to M$ be a map such that $f_*(i_n) = qz$ where $i_n \in H_n(S^n, \mathbb{Z})$ is the fundamental class and $q \neq 0$. How do I see that multiplication by $q$ annihilates $H_i(M, \mathbb{Z})$ if $1 \le i \le n - 1$?
[This argument is stolen from the end of this answer (which handles the case $q=1$).]
Let $1\leq i\leq n-1$ and $\alpha\in H_i(M,\mathbb{Z})$. By Poincare duality, there exists $\beta\in H^{n-i}(M,\mathbb{Z})$ such that $\alpha=z\cap\beta$. Since $H^{n-i}(S^n,\mathbb{Z})=0$, $f^*(\beta)=0$. Thus $0=f_*(i_n\cap f^*(\beta))=f_*(i_n)\cap\beta=qz\cap\beta=q\alpha$.