# Embeddings of vector spaces

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.

• Do you mean embeddings as topological spaces? If so, do you mean proper embeddings? My guess is the answers are Yes and No, since you mentioned the little disks operad. Jun 17 '11 at 4:16
• Smooth embeddings as manifolds. Proper I'm not sure... Jun 17 '11 at 5:46
• No not proper, just smooth. I think the answer below is what I need. Jun 17 '11 at 6:01

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).
• Thanks for the clarification. To take the last statement further, if I demand that the embeddings preserve a framing of $V$ do I get a homotopy equivalence to $F_k$ with unlabeled points, and so the highly connected as $n\to\infty$ result? Jun 18 '11 at 0:38