Lie coalgebras and rational homotopy theory II: Hopf invariants

Details Lie coalgebras and rational homotopy theory II: Hopf invariants Lie coalgebras and rational homotopy theory II: Hopf invariants

2008-09-29

arxiv.org/abs/0809.5084v5

Mathematics

Algebraic Topology

18 pages, eps and xypic figures, v5 (revised homogeneous spaces section 3.2)

Scientific paper

We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by "generalized linking/intersection invariants" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogeneous spaces (where homotopy is given by classical linking numbers), and configuration spaces (where homotopy is given by generalized linking numbers); also we propose a generalization of the Hopf invariant one question.

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