Lie coalgebras and rational homotopy theory II: Hopf invariants

Sinha, Dev; Walter, Ben

doi:10.1090/S0002-9947-2012-05654-6

Lie coalgebras and rational homotopy theory II: Hopf invariants
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by Dev Sinha and Ben Walter PDF

Trans. Amer. Math. Soc. 365 (2013), 861-883 Request permission

Abstract:

We develop a new framework which resolves the homotopy periods problem. We start with integer-valued homotopy periods defined explicitly from the classic bar construction. We then work rationally, where we use the Lie coalgebraic bar construction to get a sharp model for $\mathrm {Hom}(\pi _* X, {\mathbb Q})$ for simply connected $X$. We establish geometric interpretations of these homotopy periods, to go along with the good formal properties coming from the Koszul-Moore duality framework. We give calculations, applications, and relationships with the numerous previous approaches.

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