# How does the discrete group act on simplicial set level by level

Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set?

For example, if we have the action of $\mathbb{Z}_2$ on the topological space $S^n$ given by $t\cdot x=-x$, where $t$ is the generator of $\mathbb{Z}_2$ and $x\in S^n$, then what is the corresponding action of $\mathbb{Z}_2$ on the simplicial set$S^n$.

The simplicial structure of $S^n$ is given by $S^n_k=${ $*$} for $k< n$ and $S^n_k=\lbrace *, (i_0,i_1,\dots, i_k)\;|\; 0\leq i_0\leq\cdots\leq i_k\leq n \rbrace$ for $k\geq n$. Faces and degeneracies are given by "deleting and doubling" for catching simplicial identities.

What is the action of $\mathbb{Z}_2$ on $S^n_k$ for each $k$? When $k< n$, the action is trivial. What about the case of $k\geq n$?

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There is no reason for the action to respect the simplices in the triangulation of $S^n$ you started with. –  Mariano Suárez-Alvarez Nov 8 '10 at 4:48
(For example, suppose your simplicial set corresponds to the triangulation of the sphere given by a tetrahedron) –  Mariano Suárez-Alvarez Nov 8 '10 at 4:49
If you have a specific simplicial structure in mind, then please add that information to the queston body itself (and then remove your answer, which is not an answer!) –  Mariano Suárez-Alvarez Nov 8 '10 at 5:16
Thanks for the comment. –  Gao 2Man Nov 8 '10 at 5:26
If $G$ acts on the geometric realization $|X|$ of $X=\{X_n\}_{n\geq 1}$ then it also acts on the Eilenberg simplicial set $\mathbf{Top}(\Delta_n,|X|)$. So you're basically looking for an inclusion of simplicial sets $\varphi:X_n\to \mathbf{Top}(\Delta_n,|X|)$ which is $G$-invariant. In your example, the simplicial set you have must then be a $\mathbb{Z}_2$-orbit inside $\mathbf{Top}(\Delta,X)$. –  Somnath Basu Nov 8 '10 at 6:07