Covering with Deck group $\mathfrak{S}_3$

Question

I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be small finite complexes and the covering to be cellular. I would be happy if I have a concrete description of the attaching maps of $X$ and how $\mathfrak{S}_3$ permutes the cells of $X$.

Answer 1

Here is a picture from Topology and Groupoids

It is meant to show in (i) the Cayley graph of the presentation $\mathcal P$ of $G=S3$, $\{x,y:x^3,y^2,xyxy\}$. The Cayley graph is the $1$-skeleton of the universal cover of the cell complex $K(\mathcal P)$ of the presentation. The picture (ii) is a chosen tree in the Cayley graph. The diagram shows 3 rectangles, 2 triangles, and 3 "2-gons". Since $K$ has three 2-cells, corresponding to the relations, the universal cover will have eighteen 2-cells.

FKranhold's comment seems quite correct. A full algebraic explanation is given in Section 10.3.ii of the book Nonabelian Algebraic Topology (NAT) using free crossed modules and resolutions, and covering morphisms of groupoids. (It may seem extreme to recommend these concepts from this thick book, but these are the algebraic concepts relevant to the cellular geometry!)

If we set $r=x^3, s = y^2, t=xyxy$ in $F(x,y)$, the free group on $x,y$, the free crossed module $\delta: C(r,s,t) \to F(x,y)$, maps $r$ to $x^3$, $s$ to $y^2$, $t$ to $xyxy$, modelling the boundaries of the 2-cells of $K(\mathcal P)$ . The kernel of $\delta$ is isomorphic to the second homotopy group $\pi_2(K(\mathcal P), *)$.

The algebraic modelling procedure is also helped by using groupoids and covering morphisms of groupoids.