5
$\begingroup$

Let $G$ be a compact Lie group and $\mathcal{C}_G$ the category of $G$-spaces (ie. topological spaces endowed with continuous left $G$-actions). Is there a model category structure on $\mathcal{C}_G$ for which

(i) weak equivalences are the morphisms $f:X\rightarrow Y$ such that for all closed subgroups $H$ of $G$, $f^{H}:X^H\rightarrow Y^H$ is a weak homotopy-equivalence, and

(ii) cofibrations are the morphisms $f:X\rightarrow Y$ that have the expected $G$-homotopy extension property?

What I have read thus far strongly suggests that this is the case. Nevertheless, I would appreciate a reference that makes this explicit.

$\endgroup$
7
$\begingroup$

Yes, assuming that by "expected $G$-homotopy extension property" you mean the Serre $G$-cofibrations, not the Hurewicz $G$-cofibrations. A very explicit reference is Proposition A.1.18 in Schwede's Global homotopy theory.

$\endgroup$
7
$\begingroup$

This result has nothing special to do with compact Lie groups: it works for arbitrary topological groups $G$. And as Karol gently points out, the "expected $G$-homotopy extension property" actually means "retracts of relative $G$-cell complexes". The fibrations, like the weak equivalences are created by the fixed point functors: $p$ is a $G$-fibration iff $p^H$ is a nonequivariant Serre fibration for all closed subgroups $H$.

This is the ``Quillen model structure''; it was understood in the 1980's. There is also a classical or Hurewicz model structure whose weak equivalences are the $G$-homotopy equivalences and whose $G$-fibrations are the Hurewicz $G$-fibrations, defined by the $G$-CHP, and whose $G$-cofibrations really do have the $G$-HEP, which I imagine may be what the question had in mind. There is also a mixed model structure in which the weak equivalences are those of the Quillen model structure and the fibrations those of the Hurewicz model structure. Its cofibrations are the Hurewicz cofibrations that factor as composites of Quillen cofibrations and $G$-homotopy equivalences.

The proofs are really no different than in the nonequivariant case. Some references on my web page are http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf (Section VI.5), http://www.math.uchicago.edu/~may/PAPERS/MMMFinal.pdf (Theorem 1.8 there gives the same result as Schwede's A.1.18, with a little more detail about the properties of the Quillen model structure), and http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf (Chapter 17)

$\endgroup$
  • $\begingroup$ Indeed, in the notes I linked in my answer this is actually proven even for topological monoids and weak equivalences specified by fixed points with respect to some nice enough class of submonoids. $\endgroup$ – Karol Szumiło Aug 7 '14 at 5:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.