Homotopy Lie algebras, lower central series and the Koszul property

Abstract

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k -rescaling of the rational cohomology ring of X . Assume H * (X, Q) is a Koszul algebra. Then, the homotopy Lie algebra π * (ΩY ) ⊗ Q equals, up to k -rescaling, the graded rational Lie algebra associated to the lower central series of π 1 (X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H * (X, Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of π 1 (X) and the completion of [ΩS 2k+1 , ΩY ]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph. AMS Classification numbers Primary: 16S37, 20F14, 55Q15Secondary: 20F40, 52C35, 55P62, 57M25, 57Q45

Algebraic invariants for right-angled Artin groups

“…(7.1) Now recall that H * (X; Q) is a Koszul algebra. Therefore, by [24,Theorem B], the graded Lie algebra π * ( Y ) ⊗ Q is the q-rescaling of gr(G ) ⊗ Q, with rescaling factor of 2q. Part (1) follows from Theorem 3.4, Parts (1) and (3).…”

“…Then Y is formal. Nevertheless, the vanishing property from Part (2) and the coformality property from Part (4) fail for Y , as soon as 2q + 1 > n − 1 > 1; see [24,Example 8.5]. …”

Koszul spaces

“…Relations between the Koszul property, formality and coformality have been studied in [22] and [23], but with a more restrictive definition of Koszul algebras; their definition require Koszul algebras to be generated in cohomological degree 1. In [22,Example 4.10] the authors give an example of a space which is formal and coformal, but where the authors say the cohomology algebra is not Koszul, namely S 1 ∨ S 2 . The cohomology algebra is not Koszul according to the more restrictive definition, simply because it is not generated in cohomological degree 1, but it is Koszul according to our definition.…”

Bar complex, configuration spaces and finite type invariants for braids

“…We denote by A n [ ] the graded algebra whose degree p part is given by This type of rescaling theorem was discussed systematically in [15] in relation with the rescaling in the level of cohomology rings. A similar rescaling isomorphism for the orbit configuration space for the action on the upper half plane of a discrete subgroup of PSL(2, R) in [5].…”