A formula for topology/deformations and its significance

• Autorzy: Ruth Lawrence , Dennis Sullivan
• Języki publikacji: EN

Abstrakty

EN

The formula is
$∂e = (ad_e)b + ∑_{i=0}^{∞} (B_i)/i! (ad_e)^{i}(b-a)$,
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 coefficients are defined by
$x/(e^{x} - 1) = ∑_{n=0}^{∞} Bₙ/n! xⁿ$.
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 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.

Kategorie tematyczne

• 55P35: Loop spaces
• 16E45: Differential graded algebras and applications
• 55U15: Chain complexes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 225
• Numer: 1
• Strony: 229-242

Daty

wydano

2014

Twórcy

autor

Ruth Lawrence

• Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

autor

Dennis Sullivan

• SUNY Department of Mathematics, Stony Brook, NY 11794-3651, U.S.A.
• CUNY Graduate Center, New York, NY 10016, U.S.A.

Identyfikatory

DOI

10.4064/fm225-1-10