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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    $\begingroup$ There is no reason for the action to respect the simplices in the triangulation of $S^n$ you started with. $\endgroup$ – Mariano Suárez-Álvarez Nov 8 '10 at 4:48
  • $\begingroup$ (For example, suppose your simplicial set corresponds to the triangulation of the sphere given by a tetrahedron) $\endgroup$ – Mariano Suárez-Álvarez Nov 8 '10 at 4:49
  • $\begingroup$ 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!) $\endgroup$ – Mariano Suárez-Álvarez Nov 8 '10 at 5:16
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    $\begingroup$ 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 $\mathbf{Top}(\Delta,X)$. $\endgroup$ – Somnath Basu Nov 8 '10 at 6:07

I just noticed this from last year, and it may not be relevant now, but related to the comments above, on simplicial complexes with a certain type of action (no inversion of edges if I remember rightly) then the structure is related to Haefliger's complex of groups. (This is like the Serre theory of groups acting on trees, but here they are acting on simplicial complexes!) It is well explained in the book by Bridson and Haefliger as well as in certain papers by him.

  • $\begingroup$ I forgot to say. Give my regards to Jie Wu. $\endgroup$ – Tim Porter Dec 7 '11 at 19:56
  • $\begingroup$ Hello Tim Porter, no problem. Thanks for your answer! $\endgroup$ – Gao 2Man Dec 8 '11 at 13:50

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