For a topological space $X$ and a positive integer $k\in \mathbb{N}_{>0}$ let $F_k(X):= \{ (x_1,\ldots,x_k)\in X^k |x_i\neq x_j \text{ for } i\neq j \}$ be its $k$-configuration space. Let $f:M\to \mathbb{R}$ be a Morse-function on a compact manifold $M$. The space $GVect(f)$ of all gradient-like vector fields for $f$ is convex (if a suitable definition of gradient-like is used) and hence contractible.

Question:Is the $k$-configuration space $F_k(GVect(f))$ contractible?

My approach so far:

Let $Q_m\subset GVect(f)$ denote a set of $m\geq 0$ distinct points.

*Claim:*

- $GVect(f)$ is a manifold without boundary
- $GVect(f)\setminus Q_m$ has trivial homotopy groups
- $F_k(GVect(f))$ is a $CW$-complex.

Using parts 1 & 2 of the claim, Theorem 2.5 and the proof of Theorem 2.7 of this paper, one can show that the homotopy groups of $F_k(GVect(f))$ are trivial. Whitehead's-theorem and part 3 of the claim now imply that $F_k(GVect(f))$ is in fact contractible.

However, I'm not sure if the claim is true (or if it's even sensible) and how one may prove it.

Thank you for any contribution.