characterization of cofibrations in CW-complexes with G-action

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Is there a condition for a $G$-equivariant map $X \to Y$ to be a cofibration of $G$-spaces? Here $X$ and $Y$ are CW complexes, the group $G$ is finite, and acts by cellular maps.

I am using the model structure on CW-complexes with G-action where the fibrations and weak equivalences are those maps which are fibrations, weak equivalences respectively when we forget the $G$ action.

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There is no model structure on the category of CW-complexes since it is not complete or cocomplete, but I presume you mean the model structure on all G-spaces. – Mike Shulman Feb 20 2010 at 1:37
Since this was already on the front page, I decided to retag with model-categories. – David White Aug 18 2011 at 22:41

1 Answer

In the model structure you describe, the cofibrations should be the retracts of the free relative G-cell maps: i.e., retracts of maps obtained by attaching cells of the form $G \times S^{n} \to G \times D^{n+1}$.

One way to see this is via the following general machine: There is an adjoint pair $$G \times -: \mathbf{Top} \leftrightarrow \mathbf{GTop}: forget$$ $\mathbf{Top}$ is a cofibrantly generated model category and one can check that this adjoint pair satisfies the conditions of the standard Lemma for transporting cofibrantly generated model structures across adjoint pairs (see e.g., Hirschorn's "Model categories and their localizations" Theorem 11.3.2). Thus, it gives rise to a model structure on $\mathbf{GTop}$ such that a map in $\mathbf{GTop}$ is an equivalence(resp. fibration) iff its image under the right adjoint (forget) is so. Moreover, the generating (acyclic/)cofibrations are precisely the images under the left adjoint ($G \times -$) of the generating (acyclic/)cofibrations in $\mathbf{Top}$. This yields the description of the cofibrations as retracts of (free G-)"cellular" maps.

Also, some context:

The model structure you describe (which I'd like to call "Spaces over BG") is a localization of a model structure "G-Spaces" (where the weak equivalences are maps inducing weak equivalences on all fixed point sets). An argument along the lines of the above constructs this other model structure and identifies its cofibrations with retracts of (arbitrary) relative G-cell maps: i.e., retracts of maps obtained by attaching cells of the form $G/H \times S^n \to G/H \times D^{n+1}$ for $H$ a closed subgroup of $G$.

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