Existence of degree of smooth map between manifold and sphere

Question

I came across this statement and couldn't figure out why this is true, please help:

Let $M$ be an n-dimensional compact, connected, orientable smooth manifold without boundary. Prove that there exists a smooth map $f:M\rightarrow \mathbb{S}^n$ such that $\text{deg}(f) = k$ for arbitrary $k\in \mathbb{Z}$.

Here the degree is defined by $\int_M f^*\omega = \text{deg}(f)\cdot \int_{\mathbb{S}^n} \omega$.

Answer 1

HINT: First of all, you can produce maps $S^n\to S^n$ of arbitrary degree $k\in\Bbb Z$, so it suffices to produce a map from $M$ to $S^n$ of degree $1$. Take a coordinate ball $B\subset M$. Using a bump function appropriately to make things smooth, map a sub-ball $B'\subset B$ diffeomorphically to $S^n-\{p\}$ and $M-B'$ to $p$.