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Is it possible to replace cellular chains in the construction of Bredon homology/cohomology with cingular chains? (Possible it should be, but in that case will the Bredon homology groups be the same?)

I have strong feeling that it should be possible, but I am not sure.

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    $\begingroup$ It sounds like Illman's "Equivariant singular homology and cohomology I" (Mem. Amer. Math. Soc. 1 (1975), issue 2, no. 156) is exactly what you're looking for. I can't say anything about the possible difference in the theories, though. $\endgroup$
    – Stefan Behrens
    Commented Apr 18, 2018 at 14:31
  • $\begingroup$ Yes, I saw this paper. I am simply not sure, whether his construction (which by uniqueness of ordinary cohomology theories should be the same as Bredon, given coefficient system) is equivalent to just replacing cellular chains in the Bredon's construction with singular chains. $\endgroup$
    – Igor Sikora
    Commented Apr 18, 2018 at 14:37
  • $\begingroup$ Well, they should agree on finite complexes, since they both satisfy the relevant Eilenberg-Steenrod type axioms. But as I said, I'm not an expert on these matters. Sorry. $\endgroup$
    – Stefan Behrens
    Commented Apr 18, 2018 at 14:54

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I'll answer this question just for future references. Since (after fixing a coefficient system) the ordinary cohomology theory is unique, on G-CW-complexes cohomology defined by cellular chains will be the same as defined by singular chains. And since compactly generated G-spaces have weak homotopy type of a G-CW-complex, it is possible to use singular chains in the definition.

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