In this work, we develop a new foundation
for rational homotopy theory based on Lie coalgebras.
The work starts with a combinatorial model for the linear dual to
the Lie operad, based on a combinatorial pairing between graphs and trees
(see the paper of that name listed in the
configuration space page).
Ben Walter and I are
writing a series of three papers on this subject, the first of which
sets the basic structure (including cobracket structure, bar and cobar
constructions, and model structures) using a lift to a new category
of graph coalgebras. The second paper yields the geometric payout, developing Hopf invariants from this
point of view and uniting work of Chen, Hain, Sullivan and Boardman-Steer.
The planned third paper will remove finiteness and connectivity hypotheses.
We develop a new, intrinsic, computationally friendly approach to Lie coalgebras through graph coalgebras, which are new and likely to be of independent interest. Our graph coalgebraic approach has advantages both in finding relations between coalgebra elements and in having explicit models for linear dualities. As a result, proofs in the realm of Lie coalgebras are often simpler to give through graph coalgebras than through classical methods, and for some important statements we have only found proofs in the graph coalgebra setting. For applications, we investigate the word problem for Lie coalgebras, we revisit Harrison homology, and we unify the two standard Quillen functors between differential graded commutative algebras and Lie coalgebras.
||Lie coalgebras and rational homotopy theory, I: graph coalgebras.
(with Ben Walter)
We give a new, definitive answer to the basic question "how can cochain data determine
rational homotopy groups?" Moreover, we give a method for determining when two maps
from Sn to X are homotopic after allowing for multiplication by some integer.
We start by building integer-valued homotopy functionals from the cobar complex
on the cochains of a space, which we call generalized Hopf invariants. We show these Hopf
invariants pass to the Harrison complex, and in that setting give sharp duality
with rational homotopy. The previous paper in this series built a new framework
for understanding Lie coalgebras as quotients of graph coalgebras and applied
this to rational homotopy theory. We extend that work by giving an independent
geometric proof that our graph coalgebra models are dual to homotopy, with
cobracket dual to the Whitehead product. For applications, we investigate wedges
of spheres, homogeneous spaces, and configuration spaces; and we propose a
generalization of the Hopf invariant one question.
||Lie coalgebras and rational homotopy theory, II: Hopf invariants.
(with Ben Walter)