# Homotopy Type of Embeddings of Circles in a 3-manifold

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.

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.
• That is very right thank you : I forgot about these kind of loops. I think that what I had in mind was actually the "space of images of these embeddings", i.e the same component $Emb(C, H_2)$, but quotiented by the parametrizations of the circles, $Diff(S^1)^3$. These loops are now trivial Feb 2, 2020 at 9:22