# A formula for topology/deformations and its significance

Ruth Lawrence, Dennis Sullivan

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

## Abstract

The formula is ∂e = (ade)b + ∞σi=0 Bi/i!(ade)i(b - a); with ∂a + 1/2 [a; a] = 0 and ∂b + 1/2 [b; b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a; b; e]. The coeffcients are defined by x/ex- 1 = ∞σn=0 Bn/n!xn: The theorem is that this formula for ∂ on generators extends to a derivation of square zero on L[a; b; e]; the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the "flow" generated by e moves a to b in unit time. The immediate significance of this formula is that it computes the infinity cocommutative coalgebra structure on the chains of the closed interval. It may be derived and proved using the geometrical idea of at connections and one-parameter groups or ows of gauge transformations. The deeper significance of such general DGLAs which want to combine deformation theory and rational homotopy theory is proposed as a research problem.

Original language American English 229-242 14 Fundamenta Mathematicae 225 1 https://doi.org/10.4064/fm225-1-10 Published - 2014
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