Is any map homotopic to a map having fixed point?

Question

I am familiar with the result that any continuous map $ f : S^{n} \rightarrow S^{n} $ is homotopic to a map having fixed point . Now I want to know whether it's true in general . Suppose we have a simply connected smooth manifold M of dimension m and a continuous map $ f : M \rightarrow M $. Is it possible to construct a continuous map $ g : M \rightarrow M $ which has fixed point and is homotopic to f ? What about the case when M is connected but not simply connected ?

Answer 1

Yes, this is true, and you only need $M$ to be connected. It follows quite easily from the well-known

Homogeneity Lemma. [Milnor, Topology from the Differentiable Viewpoint, p22] Let $y$ and $z$ be arbitrary interior points of the smooth, connected manifold $M$. Then there exists a diffeomorphism $h: N \to N$ that is smoothly isotopic to the identity and carries $y$ into $z.$

Thus if $f : M\to M$ is smooth, we can take any $z \in M$ and choose $y = f(z),$ so that the resulting $h$ satisfies $h \circ f(z)=z.$ Since $h$ is isotopic to the identity, $h \circ f$ is isotopic to $f,$ so we're done.

If $f : M \to M$ is merely continuous, then to apply this argument you also need to know that every continuous map is homotopic to a smooth map, which follows from the Whitney approximation theorem.

Since you asked about continuous maps and not smooth ones: you should be able to emulate this argument without the smooth structure (and thus without needing the approximation), since connected topological manifolds are also homogeneous.