PROBLEM: For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic ,or even of degree 1
The following is my idea: First, choosing an arbitrary open coordinate ball $B$ on $M$ and then collapsing $M\setminus B$ to a single point gives a continuous map $f:M \to S^n$.Also, we may assume all $M\setminus B$ is mapped to the north pole $\mathbb N$.
It is known that any continuous map $f$ between two manifolds $M_1$ and $M_2$ is homotopic to a smooth map $F$.
Moreover, can we require $|F-f|\le \epsilon$ for any positive continuous function $\epsilon$ on $M_1$ ?
Here the norm $|\cdot|$ can be viewed as a Euclidean norm when we embeds $M_2$ to some $\mathbb R^m$
Personally, I believe it's true. If the statement above is indeed true, then we denote $F$ the smooth map homotopic to $f$ and choose $\epsilon$ small enough. Also, we choose a regular value $\mathbb P$ near the South pole $\mathbb S$ of $S^n$. Then, $\mathbb P$ can only has one preimage and thus the degree of $F$ must be 1 or -1