A formula for topology/deformations and its significance

Ruth Lawrence and Dennis Sullivan

Abstract: The formula is

\d{e}=ad_e(b)+\sum_{i=0}^\infty{B_i\over{}i!}(ad_e)^i(b-a)

with \d{a}+\half[a,a]=0 and \d{b}+\half[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 coefficients are defined by x/(e^x -1)=\sum_{n=0}^\infty{B_n\over{}n!}x^n. The theorem is that (1) this formula for \d on generators extends to a derivation of square zero on L[a,b,e]; (2) the formula for \d{e} is unique satisfying the first property, once given the formulae for \d{a},\d{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 flat connections and one-parameter groups or flows 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.

AMS subject classification: 55U15 16E45 55P35

Length: 14 pages

Reference: Fundamenta Mathematica 225 (2014) 229-242

MR3205571 (review by Andrey Yu. Lazarev .)

You can download this paper from here . The previous version of the paper was entitled A free differential Lie algebra for the interval and can be found at arXiv:math.AT/0610949.

Last updated April 15th, 2018.
ruthel@math.huji.ac.il