Homotopy Type of Embeddings of Circles in a 3-manifold

Question

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.

https://ibb.co/rQt0dg7

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.

Answer 1

The space of embeddings of even one of these circles is nontrivial. The manifold $H_2$ deformation retracts to $S^1 \vee S^1$. If a loop of embeddings $f_t:S^1 \to H_2$ were contractible, then certainly so would be the loop of maps $f_t:S^1 \to S^1 \vee S^1$. But if $f_0:S^1 \to S_1 \vee S^1$ is the inclusion of one of the two circle factors, then $f_t(\theta)=f_0(\theta+t)$ for $t \in [0,2\pi]$ is certainly not a contractible loop as can be seen from typical covering space theory.

In response to your comment: if you quotient by reparametrizations, then the space of embeddings has trivial $\pi_1$ for the same reason. Homotopy classes of maps into $H_2$ are the same as homotopy classes of maps into $S^1 \vee S^1$. Any map $S^1 \to S^1 \vee S^1$ lifts to a map on the universal covers $\mathbb{R} \to T$ where $T$ is the regular 4-valent tree. Homotopy on the base space level lifts to homotopy on the universal covers, and quotienting by reparametrization is effectively asking that the integral points in $\mathbb{R}$ remain fixed at the vertices in $T$. Now your loop of maps $S^1 \to S^1 \vee S^1$ mod reparametrization is the same as a loop of maps $[0,1] \to T$ rel boundary. This can be viewed as a map $S^2 \to T$, and we know that all such maps are homotopic. In particular, the loop of maps $[0,1] \to T$ homotopic to a constant sequence of such maps.