On spineless cacti, Deligne’s conjecture and Connes–Kreimer’s Hopf algebra

Abstract

Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.From this description one obtains several useful corollaries [18]. The ones relevant to the present discussion are listed below. Corollary 2.16The quasi-operad of normalized spineless cacti is homotopy associative and thus its homology quasi-operad is an operad.Moreover as quasi-operads normalized spineless cacti and spineless cacti are homotopy equivalent via a homotopy of quasi-operads. And finally:Corollary 2.17 Normalized spineless cacti are operadically quasi-isomorphic spineless cacti. I.e. their homology operads are isomorphic.

Dimension vs. genus: A surface realization of the little k-cubes and an E∞operad

“…We do not wish to go into the gory details. The proof is a straightforward adaption from that of Cacti presented in [Ka1,Ka2]. For the semi-direct product the first homeomorphism is given by reading off the weights at each boundary and then taking the projective class of the weights at each boundary individually.…”

“…This is not surprising, but here we have a very geometric picture. For the actions, we can take the action of the chain operad on itself of the action on the Hochschild complex as defined in [Ka2]. Using the former action, we obtain a universal geometric point of view.…”

“…In [Ka2] (see also [K3]) we introduced a dual graph for quasi-filling arc graphs. Here we extend this notion to all graphs.…”

The motivic Thom isomorphism

“…[Kontsevich has suggested that this is always so, but his program of proof fails, by the arguments of [4], and at present the question seems to be open.] Connes and Kreimer [12, 21, There are deep connections between the Grothendieck -Teichmüller group and the Lie algebras of these automorphism groups [29,54], and it seems likely that they (and the theory of quasisymmetric functions, via free Lie algebras) will eventually be understood to be intimately related; the appearance of polyzeta values in the theory of quantum knot invariants (cf. eg [36]) is another source of recent interest in this subject.…”

From Physics to Number Theory via Noncommutative Geometry

“…The CK Hopf algebra also appears in relation to a conjecture of Deligne on the existence of an action of a chain model of the little disks operad on the Hochschild cochains of an associative algebra (cf. Kaufmann [75]). …”