Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, i.e. the induced map $f_*: H_n\left(M\right) \to H_n\left(S^n\right)$ sends the fundamental class $\left[M\right]$ to $\left[S^n\right]$. My idea was the following:
Consider an arbitrary point $x \in M$. Then take a chart $\phi: U_x \xrightarrow{\approx} \mathbb{R}^n$ that maps $x$ to a nonzero value $\phi(x)$. Then normalize this value to $\frac{\phi(x)}{\Vert \phi(x) \Vert}$. We define our map $f: M \to S^n$ to be exactly this map. Does this construction work? I think from here on I'd have a solution to the exercise but I don't know if this map $f$ is valid.
Best Answer
My favorite method of constructing a degree $1$ map is to take an embedded open ball in $M$ and crush its complement to a point.