# Iterated higher Whitehead products and L-infinity structure for Davis—Januszkiewicz space

# Iterated higher Whitehead products and L-infinity structure for Davis—Januszkiewicz space

**Zoom link: https://princeton.zoom.us/j/92116764865**

**Passcode: 114700**

In this talk I discuss higher Whitehead products, invariants in unstable homotopy theory, which are considered in the context of the studying Davis—Januszkiewicz spaces and moment-angle complexes.

It is known that rational homotopy groups of loop space form the homotopy Lie algebra in which the Jacobi identity holds. There is a structure of L-infinity algebra, the generalization of Lie algebra for which we have n-ary brackets that satisfy the generalized Jacobi identities. In general, we do not know, what relations hold for (canonical) higher Whitehead products.

There is a minimal simplicial complex K = \partial \Delta(\partial \Delta(1,2,3), 4,5), for which iterated Whitehead products [[mu_1, mu_2, mu_3], mu_4, mu_5] is defined. In this talk I represent the full description of Pontryagin algebra and homotopy Lie algebra of Davis—Januszkiewicz space for this K, using Whitehead products. We will see that relations on Whitehead products have the form of L-infinity identities. I also represent Adams-Hilton models for Davis—Januszkiewicz space (for arbitrary simplicial complex), which play an important role in obtaining these relations.