I recall vaguely once reading that a *cell complex*—constructed like a CW-complex but without assuming the cells are appended in order of increasing dimension —"is" actually a CW-complex. I cannot remember if this means there is a homotopy equivalent CW-complex or something stronger.


Is there a similar result for $G$-cell complexes vs. $G$-CW-complexes, for $G$ a compact Lie group?



Is a $G$-cell complex always a $G$-CW complex?
As Najib says in the comments to the question, the proof of the classical statement can be easily-ish adapted to the equivariant case. Let's see the details


**Lemma** Let $X$ be a $G$-CW-complex and let $f:G/H\times S^n\to X$ be a $G$-equivariant continuous map. Then $f$ factors up to equivariant homotopy through the $n$-skeleton of $X$


*Proof:* This is just a special case of the cellular approximation theorem (theorem I.3.4 in May's *Equivariant Homotopy and Cohomology Theory*).


A $G$-equivariant continuous map from $G/H\times S^n$ is the same thing as a continuous map from $S^n$ to $X^H$ and the same hold for equivariant homotopies. Hence we need to show that, if $X^{(n)}$ is the $n$-skeleton of $X$, the inclusion $(X^{(n)})^H\to X^H$ is $n$-connected. But the cofiber $X^H/(X^{(n)})^H\cong (X/X^{(n)})^H$ is obtained by adding cells of the form $(G/K\times D^m)^H$ for $m>n$, and so it is $n$-connected. $\square$


Armed with the lemma, we can prove that every $G$-complex $Y$ is homotopy equivalent to a $G$-CW-complex. I will prove only the case where $Y$ has finitely many cells, the general case just needs a transfinite induction that I don't want to write down right now.


We will proceed by induction on the number of cells. If $Y$ has only one cell, then it has of course the structure of a $G$-CW-complex, obtained by using a CW structure on the disc.


Suppose now that the thesis is true for all $G$-complexes with $m$ cells. Then, if $Y$ is a $G$-complex with $(m+1)$ cells, we can write it as the pushout $Y=Y'\amalg\_{G/H\times S^n} G/H\times D^{n+1}$ for some $H$ and $n$. But, by the previous lemma, the attaching map $G/H\times S^n\to Y'$ factors up to homotopy through the $n$-skeleton of $Y'$. So we can attach it together with the other $(n+1)$-cells without changing the homotopy type of $Y$. Hence $Y$ has the homotopy type of a $G$-CW-complex.


