The ranks of the homotopy groups of a space of finite LS category

Félix, Yves; Halperin, Stephen; Thomas, Jean‐Claude

doi:10.1016/j.exmath.2006.04.002

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L-equivalence1

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2009

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“…Lemma 5.8 then implies that I.i C 1/ contains an infinite dimensional Lie subalgebra of depth 1. Finally according to [7] each infinite dimensional Lie subalgebra of depth 1 contains a free Lie algebra on two generators.…”

Section: L-equivalencementioning

confidence: 99%

The structure of homotopy Lie algebras

Félix¹,

Halperin²,

Thomas³

2009

Comment. Math. Helv.

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In this paper we consider a graded Lie algebra, L, of finite depth m, and study the interplay between the depth of L and the growth of the integers dim L i. A subspace W in a graded vector space V is called full if for some integers d , N , q, dim V k Ä d P kCq iDk dim W i , i N. We define an equivalence relation on the subspaces of V by U W if U and W are full in U C W. Two subspaces V , W in L are then called L-equivalent (V L W) if for all ideals K L, V \ K W \ K. Then our main result asserts that the set L of L-equivalence classes of ideals in L is a distributive lattice with at most 2 m elements. To establish this we show that for each ideal I there is a Lie subalgebra E L such that E \ I D 0, E˚I

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Section: L-equivalencementioning

confidence: 99%