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Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of Gerstenhaber algebra (with some degree shift), and to $H_\bullet(LM,M)$ the structure of Gerstenhaber coalgebra. See "On String Topology Operations and Algebraic Structures on Hochschild Complexes" by Manuel Rivera for a nice exposition.

We can read off the Gerstenhaber algebra structure via the Burghelea-Fiedorowicz-Goodwillie isomorphism $\mathrm{HH}_\bullet(C_\bullet(\Omega M))\cong H_\bullet(LM)$ combined with the Poincaré duality, which is in fact a BV isomorphism. For the (original) reference, see Eric J. Malm's "String topology and the based loop space".

My questions are:

(1) How can we obtain the Gerstenhaber (or BV-) coalgebra structure in a cohomological manner just like the above?

(2) In general, what kind of structure does give us a Gerstenhaber coalgebra?


If $M$ is simply connected, Theorem 22 (ii) of Rivera's paper seems to give a partial answer to (1): the coalgebra structure is recovered from the one on the co-Hochschild homology of some space. This is based on a discussion in the same paper (and gives a partial answer to (2)): it requires a DG Frobenius (commutative and co-commutative) algebra to obtain a Gerstenhaber coalgebra structure (Theorem 17), which I haven't totally understood yet. Anyway, being simply connected invokes plenty of tools from rational homotopy theory to work with.

However, my interest is in the $K(\pi,1)$ case, and I have no idea how to obtain Frobenius algebras (or something similar) for the co-Hochschild homology to get a Gerstenhaber coalgebra.

Any ideas or comments are appreciated.

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  • $\begingroup$ Hi, take a look at the introduction here: arxiv.org/abs/2308.09684. This is an explicit algebraic model for the coproduct in the non-simply connected case. This may be reformulated in terms of the coHochschild complex of an appropriate coalgebra model for M. In the non-simply connected setting, chain level Poincaré duality (with local coefficients) can formulated in the language of comodules using Koszul duality theory. The cobracket requires a bit of more work but it can be worked out as well. Send me an email if you have more questions! $\endgroup$
    – Manuel Rivera
    Commented Sep 2 at 1:43
  • $\begingroup$ @ManuelRivera Thank you for the comment. I'll take a look at your preprint. $\endgroup$
    – Qwert Otto
    Commented Sep 3 at 5:41
  • $\begingroup$ Namely, instead of using a Frobenius algebra structure, in the non-simply connected case you may express Poincaré duality (capping with the fundamental class of $M$) as inducing a Calabi-Yau structure $A^! \simeq A$ for $A=C_*(\Omega M)$. This can be expressed entirely in the language of chains and comodules using that $A$ is quasi-isomorphic to the cobar construction on the singular chains $C_*(M)$. Then you plug $C_*(M)$ into the coHochschild complex to obtain a model for the free loop space and the use the appropriate version of duality on $C_*(M)$ to recover string topology. $\endgroup$
    – Manuel Rivera
    Commented Sep 3 at 17:56

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