Twisting of properads
Sergei Merkulov

TL;DR
This paper investigates the twisting endofunctor on dg properads related to Lie bialgebras, revealing new structures and cohomological properties, with implications for cyclic homotopy associative algebras and string topology.
Contribution
It introduces a new twisting endofunctor Tw, analyzes its properties, and connects it to the homotopy theory of Lie bialgebras and applications in string topology.
Findings
Tw(P) becomes a properad under quasi-Lie bialgebras.
The cohomology of Tw(Lieb) contains the haired graph complex cohomology.
Tw(Holieb) is quasi-isomorphic to Lieb, linking to homotopy theory of triangular Lie bialgebras.
Abstract
We study Thomas Willwacher's twisting endofunctor tw in the category of dg properads P under the operad of (strongly homotopy) Lie algebras. It is proven that if P is a properad under properad Lieb of Lie bialgebras , then the associated twisted properad tw(P) becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad tw(Lieb) is highly non-trivial -- it contains the cohomology of the so called haired graph complex introduced and studied recently in the context of the theory of long knots and the theory of moduli spaces of algebraic curves. Using a polydifferential functor from the category of props to the category of operads, we introduce the notion of a…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Algebraic structures and combinatorial models
