Let $G$ be a group (not a topological group, just a group). By a $G$-complex I mean a CW-complex with an action of $G$ that takes cells to cells so that the pointwise and setwise stabilizer of each cell coincide. Does anyone know a good reference for the fact that every $G$-complex is $G$-homotopy equivalent to a $G$-simplicial complex of the same dimension?
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$\begingroup$ Can we make any assumptions at all about $G$? Most of the literature proves results when $G$ is finite. $\endgroup$– David WhiteCommented Feb 8 at 20:26
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$\begingroup$ I am interested in the general case although a reference that covers the case when $G$ is finite could starting point. $\endgroup$– Peter KrophollerCommented Feb 8 at 20:47
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1$\begingroup$ Google supplies this ams.org/journals/tran/1980-258-02/S0002-9947-1980-0558178-7/… Lemma 4.3 which seems related $\endgroup$– Benjamin SteinbergCommented Feb 8 at 20:53
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1$\begingroup$ A reference for the equivariant Whitehead theorem outside the compact Lie case is in Matumoto's "On G-CW complexes and a theorem of JHC Whitehead" (ncatlab.org/nlab/files/matumoto.pdf). This at least reduces the question to one about constructing a weakly equivalent simplicial complex which seems roughly the same as section 5 of Waner's paper. The "equal dimension" part may be a little trickier. Unfortunately this doesn't really count as a reference. $\endgroup$– Tyler LawsonCommented Feb 8 at 22:29
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1$\begingroup$ @TylerLawson, Waner’s paper doesn’t assume compactness footprint the equivariant whitehead theorem but unfortunately does in section 4, which I missed when I gave the reference. $\endgroup$– Benjamin SteinbergCommented Feb 9 at 2:54
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