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Suppose I have a group $G$ which acts on the homology $H_*(C)$ of a differential graded vector space $C$. Can I always lift this to a homotopy coherent $G$-action on $C$?

My first naive thought was that $H_*(C)$ is homotopy equivalent to $C$, so I can transport the action on $H_*(C)$ to a homotopy coherent action on $C$. Will that give me a lift?

What can be said about the uniqueness of such lifts?

Thanks for any hints.

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Yes, you can always lift. (Assuming you work over a field, otherwise, I don't know.)

Note that an action of $G$ is the same as an action of the group algebra $k[G]$ where $k$ is your base field. You have a quadratic-linear (inhomogeneous) presentation of $k[G]$, with generators $g \in G$ and relations $g \cdot h = gh$. It's easy to check that this defines a Koszul algebra (the Koszul dual is the bar construction of $B(k[G])$).

Choosing representatives for homology classes, you can find a deformation retract of $C$ onto $H_*(C)$ as usual. You can then apply the Homotopy Transfer Theorem (see e.g. Theorem 47 of the paper Homotopy BV-algebras by Gálvez-Carrillo, Tonks, and Vallette for a statement in the context of inhomogeneous operads, which specializes to inhomogeneous algebras) to transfer the $k[G]$-action on $H_*(C)$ to an $\Omega(B(k[G]))$-action on $C$. Moreover $C$ and $H_*(C)$ are quasi-isomorphic seen as $\Omega(B(k[G]))$-modules. Since everything is concentrated in arity 1, you can view your operad as a non-symmetric one, so I don't think you need the characteristic zero assumption on $k$.

As for unicity, I am not sure. Yalin's paper Realization Spaces of Algebraic Structures on Cochains contains a way to compute the connected components of the space of all possible lifts of homotopy-coherent structures on $C$ given the action on $H_*(C)$. It should apply here.

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  • $\begingroup$ Over a field the chain complex is quasi-isomorphic to its cohomology... $\endgroup$
    – Denis Nardin
    Commented Feb 6, 2019 at 11:06
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    $\begingroup$ @DenisNardin it appears that the question itself speaks of "vector spaces" so presumably it is not answerer's fault that they assume that everything is over a field $\endgroup$
    – rori
    Commented Feb 6, 2019 at 11:10
  • $\begingroup$ Thank you, Naijib. Just to be sure that I am following your argument correctly: You mention that the group algebra is Koszul because that implies that the corresponding operad (being non-trivial only in arity 1) is Koszul such that you can apply that HT theorem? Another small question: I was also briefly thinking about applying HTT here, but it seemed a bit of an overkill. In the question I sketched I very naive way of transferring which should work in this special case. Do you think that would also work? $\endgroup$
    – Lukas Woike
    Commented Feb 6, 2019 at 11:41
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    $\begingroup$ @Lukas Yes, if an algebra $A$ is Koszul then the operad $(0,A,0,0,\dots)$ is Koszul if I'm not mistaken. The HTT is probably overkill here, but I think that if you write down the details of the strategy from your question, you essentially end up reproving HTT for $A$-modules, no? $\endgroup$
    – Najib Idrissi
    Commented Feb 6, 2019 at 12:21
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    $\begingroup$ @NajibIdrissi I probably interpreted "homotopy coherent action" in a different way than you: I interpreted it as a map of $E_1$-spaces $G→\mathrm{Map}(Y,Y)$ where $\mathrm{Map}$ is the derived mapping space of $Y$. Then since $\mathrm{Map}(Y,Y)\cong \mathrm{Map}(X,X)$ as $E_1$-spaces we are done. $\endgroup$
    – Denis Nardin
    Commented Feb 6, 2019 at 12:25

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