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.



Homotopy coherent action on differential graded vector space
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.


