The Stasheff model of a simply-connected manifold and the string bracket

Dual Hodge decompositions and derived Poisson brackets

We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov. We give a topological interpretation of such Lie Hodge decompositions in terms of $S^1$-equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the H…

L-S categories of simply-connected compact simple Lie groups of low rank

We determine the L-S category of Sp(3) by showing that the 5-fold reduced diagonal $\widebarΔ_5$ is given by $ν^2$, using a Toda bracket and a generalised cohomology theory $h^*$ given by $h^*(X,A) = \{X/A,{\mathbb S}[0,2]\}$, where ${\mathbb S}[0,2]$ is the 3-stage Postnikov piece of the sphere spectrum ${\mathbb S}$. This method also yields a general result that $\cat(Sp(n)) \geq n+2$ for $n \geq 3$, which improves the result of Singhof.

弦括号到环路乘积的归约

A reduction of the string bracket to the loop product

本文证明了在流形BV exact的前提下，Chas-Sullivan弦括号可归约为环路乘积与BV算子的复合，并将该结论推广至连通紧Lie群分类空间，同时探讨了相关空间的BV exactness及弦括号非平凡性。

The Yamabe invariant for non-simply connected manifolds

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is non-negative for all closed simply connected manifolds of dimension $\ge 5$. We extend this to show that Yamabe invariant is non-negative for all closed manifolds of dimension $\ge 5$ with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed manifolds of dimension $\ge 5$…

Hodge decomposition of string topology

Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ of the free loop space of $X$ preserves the Hodge decomposition of $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in …