Is there always a homotopy taking a point to another in a connected manifold? Let $M$ be a connected topological $n$-manifold. Note this implies $M$ is path-connected.
Let $a,b \in M$. Must there always exist a continuous $H:M \times [0,1] \to M$ such that $H(m,0)=m$ for all $m \in M$, and $H(a,1)=b$?
We can note that this is true for $\mathbb{R}^n$, by considering $(x,t) \mapsto x+t(b-a)$, and it is also true for $S^1$ by considering $(e^{i\theta}, t)\mapsto e^{i(\theta+t(\theta_b-\theta_a))}$ where $a=e^{i\theta_{a}}, b=e^{i\theta_b}$.

For manifolds with boundary: no, you can't take boundary points to non-boundary points.
For manifolds without boundary: yes. For 1-manifolds, this is easy. For $n > 1$, take a neighbourhood of a (shortest) path between your points. After removing problems (by taking a subset), that's homeomorphic to a disk, with both points in the interior. Take any homeomorphism of the disk that interchanges the points while fixing the boundary, and pass it through the homeomorphism to the neighbourhood of the path.
