Set of proper homotopy classes of arcs in a manifold

Question

Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that $f^{-1}(\partial M) = \{ 0,1\}$ and such that $f(0),f(1) \in \partial_0 M$. Actually, I am interested in this set considered up to homotopies satisfying the same conditions. For the sake of having a name for it, I'll call it $\text{Arc}(M, \partial_0 M)$.

Now, pick a base point $\ast$ in $\partial_0 M$. There is a map $w : \text{Arc}(M, \partial_0 M) \to \pi_1(\partial_0 M) \backslash \pi_1(M) / \pi_1(\partial_0 M)$ given by taking whiskers from $\ast$ to the two feet $f(0)$ and $f(1)$ and pre/post-composing $f$ with these whiskers.

Is this map a bijection? Any references are welcome since I'd imagine this is well known if true.

Answer 1

Yes, this map is a bijection.

The condition $f^{-1}(\partial M)=\{0,1\}$ is superfluous. Using a collar neighborhood on $\partial M$ you can show that the space of continuous maps that you defined is a deformation retract of the space of maps $f\colon [0, 1]\to M$ that satisfy just the condition $f(0), f(1)\in \partial_0 M$.

Let us denote the space of such maps by $(M, \partial_0 M)^{(I, \partial I)}$. It is the homotopy pullback of the diagram

$$ \partial_0 M \rightarrow M \leftarrow \partial_0 M.$$

You are asking about $$\pi_0 \left((M, \partial_0 M)^{(I, \partial I)}\right).$$ The homotopy pullback square gives rise to a long exact sequence in homotopy. The relevant section of the LES is the following (implicitly choosing a basepoint in $\partial_0 M$)

$$ \pi_1(\partial_0 M)\times \pi_1(\partial_0 M) \to \pi_1(M) \to \pi_0\left((M, \partial_0 M)^{(I, \partial I)}\right)\to \pi_0(\partial_0 M)\times \pi_0(\partial_0 M)=\{1\}.$$

This is not an exact sequence of groups. Rather, there is an action of $ \pi_1(\partial_0 M)\times \pi_1(\partial_0 M)$ on $\pi_1(M)$, where the two copies of $ \pi_1(\partial_0 M)$ act on the left and on the right. $\pi_0\left((M, \partial_0 M)^{(I, \partial I)}\right)$ is the quotient set of $\pi_1(M)$ by this action.