HOMOTOPY LIE-ALGEBRAS AND SUBMANIFOLDS

[摘要] Let V be a closed oriented connected manifold of dimension n + q and let G be a closed connected subgroup of SO(q). We consider the homotopy Lie algebra of V, i.e. the graded abelian group consisting of the homotopy groups of the loop space of V, equipped with the Samelson Lie bracket. We explore a new relationship between the structure of the rationalized homotopy Lie algebra of V and the set of closed oriented codimension-q submanifolds W of V having G as normal structure group. Main result: If the rank of the periods of the Poincare: transfer of the normal characteristic classes of some submanifold W as above is greater than two, then the rational homotopy Lie algebra of V contains a free graded Lie algebra on two generators. For G = SO(q) one obtains as a corollary new restrictions on the homological behaviour of the embeddings of a given W in a given elliptic formal manifold V (e.g. V = homogenous space of positive Euler characteristic), formulated in terms of the cohomology algebras and the Pontrjagin classes of the two given manifolds.

[发布日期] 1994-01-17