A formula for topology/deformations and its significance

Abstract

The formula is $\partial{e}=({\rm ad}_e)b+\sum_{i=0}^\infty{\frac{B_i}{i!}}({\rm ad}_e)^i(b-a)\>,$ with $\partial{a}+{1\over2}[a,a] =0$ and $\partial{b}+{1\over2}[b,b] =0$, 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 ${x\over{e^x-1}}=\sum_{n=0}^\infty{B_n\over{}n!}x^n$. The theorem is that (I) this formula for $\partial$ on generators extends to a derivation of square zero on $L[a,b,e]$, (II) the formula for $\partial{e}$ is unique satisfying the first property, once given the formulae for $\partial{a}$ and $\partial{b}$, 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.