Let $V$ be an $n$-dimensional vector space. Is the space of embeddings
$$
\coprod\_1^{k} V \to V
$$
path connected for large enough $n$? Clearly $n=1$ is not enough, but I feel like $n=2$ is enough for $1$-connected. Does the space become highly connected as $n\to \infty$? This feels like it is equivalent to a question about the little disks operads, but I don't know how to frame it as such.



Embeddings of vector spaces
No, it is not connected: for example, if $k=1$ it has two path components, given by the two orientations with which $V$ can be embedded into itself.


In general, it has the homotopy type of $F\_k(V; O(n))$ the space of configurations of $k$ particles in $V$ with labels on the orthogonal group, which has $2^k$ path components given by the possible configurations of the orientations.


If you ask for the embedding of each $V$ to be orientation-preserving, then the space is path-connected for $n > 1$ by Tilman's argument (as $SO(n)$ is connected).


