A formula for topology/deformations and its significance

作者：Lawrence Ruth^*; Sullivan Dennis

来源：Fundamenta Mathematicae, 2014, 225(1): 229-242.

DOI：10.4064/fm225-1-10

• rational homotopy theory
• infinity structure
• deformation theory

The formula is
partial derivative e = (ad(e))b + Sigma(infinity)(i=0) Bi/i!(ad(e))(i)(b-a),
with partial derivative a + 1/2[a, a] = 0 and partial derivative 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 coefficients are defined by
x/e(x) - 1 = Sigma(infinity)(n=0) Bn/n! x(n).
The theorem is that
this formula for partial derivative on generators extends to a derivation of square zero on L[a, b, e];
the formula for partial derivative(e) is unique satisfying the first property, once given the formulae for partial derivative(a) and partial derivative(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.

• 出版日期2014