I am wondering how one could compute the homotopy type (or weak homotopy type) of the embedding space which I describe now. I suspect it to be contractible but this is merely an intuition.
Consider a 3-dimensional 2-handlebody $H_2$ and three copies of the circle $C= S^1 \amalg S^1 \amalg S^1$. I am interested in the component of $Emb(C, H_2)$ containing the embeddings where the first circle "wraps" once around the first genus of $H_2$, the second circle "wraps" once around the second genus and the third copy of $S^1$ cirles them both as illustrated in the picture in the link below.
It is known that such spaces have trivial homotopy groups of degree $> 1$ so the fundamental group remains to be computed. Any insight is welcome ! Cheers.