CoHochschild homology of chain coalgebras
| dc.creator | Hess, Kathryn | |
| dc.creator | Parent, Paul-Eugene | |
| dc.creator | Scott, Jonathan | |
| dc.date | 2007-11-07 | |
| dc.date | 2008-07-15 | |
| dc.date.accessioned | 2026-07-07T09:49:56Z | |
| dc.date.available | 2026-07-07T09:49:56Z | |
| dc.description | Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on $\cohoch (C_{*}K)$ when $K$ is a simplicial suspension. The coHochschild complex construction is topologically relevant. Given two simplicial maps $g,h:K\to L$, where $K$ and $L$ are reduced, the homology of the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of $g$ and $h$, and this isomorphism respects comultiplicative structure. In particular, there a isomorphism, respecting comultiplicative structure, from the homology of $\cohoch(C_{*}K)$ to $H_{*}\op L|K|$, the homology of the free loops on the geometric realization of $K$. | |
| dc.description | 30 pages; some minor structural changes, new explicit formulas for comultiplicative structure in the case of suspensions; final version, to appear in JPAA | |
| dc.identifier | https://arxiv.org/abs/0711.1023 | |
| dc.identifier | http://arxiv.org/abs/0711.1023 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/164778 | |
| dc.subject | Algebraic Topology | |
| dc.subject | 16E40, 19D55 (Primary); 18G60, 55M20, 55U10, 81T30 (Secondary) | |
| dc.title | CoHochschild homology of chain coalgebras | |
| dc.type | text |