Contractibility of a configuration space 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.
 A: It seems that $GVect(f)$ is a manifold without boundary. 
Building on my answer to your last question (did you prove all of it?) let us argue as follows:
A vector field $X$ is in 
$GVect(f)$ if:


*

*$X(p)=0$ for each critical point $p$ of $f$. This describes a closed linear subspace $G_1$ of the Frechet space $\mathfrak X(M)$.

*Near each critical point $p$ the function $df(X)$ is Morse with a maximum at $p$. This is a $C^2$-open condition in $G_1$: The differential must be transversal to the zero section, at $p$.

*Off the critical points we have $df(X)<0$. This would be a $C^1$-open condition in $G_1$ if it holds on a closed subset of $M$. We can take the closed subset as the complement of the union of small open neighborhoods of the critical points of $f$. But these neighborhoods depend on $X$. So we have to look at all of them and take the union. Since the union of open sets is open, we are done. 
So $GVect(f)$ is open in the Frechet space $G_1$. 
The rest seems to be done by the comments to your question, by Geoffroy.
