Regarding 1: The Euler characteristic of a compact manifold M is the self-intersection of the diagonal $\Delta\subset M\times M$" />. If F is homotopic to the identity, then the diagonal (the graph of the identity) can be deformed to the graph of F. If F has no fixed points, this graph is disjoint from the diagonal; so the self-intersection, and thus the Euler characteristic, is 0.

Note that for any M, the product $M\times\mathbb{R}$" /> admits such an F, so the condition that M be compact is necessary.

Regarding 1: The Euler characteristic of a compact manifold M is the self-intersection of the diagonal $\Delta\subset M\times M$. If F is homotopic to the identity, then the diagonal (the graph of the identity) can be deformed to the graph of F. If F has no fixed points, this graph is disjoint from the diagonal; so the self-intersection, and thus the Euler characteristic, is 0.

Note that for any M, the product $M\times\mathbb{R}$ admits such an F, so the condition that M be compact is necessary.