Define what it means for a manifold to be oriented, and define a volume form on an oriented manifold.

Prove carefully that, for a closed connected oriented manifold of dimension $n$, $H^{n}(M)=\mathbb{R}$.

[You may assume the existence of volume forms on an oriented manifold.]

If $M$ and $N$ are closed, connected, oriented manifolds of the same dimension, define the degree of a map $f: M \rightarrow N$.

If $f$ has degree $d>1$ and $y \in N$, can $f^{-1}(y)$ be

(i) infinite? (ii) a single point? (iii) empty?

Briefly justify your answers.