A survey on homological perturbation theory

Johannes Huebschmann

Description

A survey on homological perturbation theory J. Huebschmann USTL, UFR de Mathematiques CNRS-UMR 8524 59655 Villeneuve d'Ascq Cedex, France Gottingen, September 14, 2010 1

• hpt


• steenrod ?i-products


• perturbation theory


• algebraic struc- ture


• products invariants


• homotopy than strict


• higher homotopies


• rational function

Sujets

• Kodaira
• Perturbation theory
• Rational function

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 189
• Langue: English

Extrait

A

survey on homological perturbation theory

J. Huebschmann

USTL,UFRdeMath´ematiques CNRS-UMR 8524 59655Villeneuved’AscqC´edex,France [e-mail]

G¨ottingen,September14,2010

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Abstract Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, is nowa- days a standard tool to handle algebraic incarnations of higher homotopies. A basic observation is that higher homotopy struc- tures behave much better relative to homotopy than strict struc- tures, and HPT enables one to exploit this observation in various concrete situations. In particular, this leads to the effective cal- culation of various invariants which are otherwise intractable. Higher homotopies and HPT-constructions abound but they are rarely recognized explicitly and their significance is hardly understood; at times, their appearance might at first glance even come as a surprise, for example in the Kodaira-Spencer approach to deformations of complex manifolds or in the theory of folia- tions. The talk will illustrate, with a special emphasis on the compat- ibility of perturbations with algebraic structure, how HPT may be successfully applied to various mathematical problems aris- ing in group cohomology, algebraicK-theory, and deformation theory.

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Contents

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Origins of homotopy and higher homo- topies 4

Some 20’th century higher homotopies 5

Homological perturbations — HPT 7

Perturbation theory for chain equiva- lences 11

Iterative perturbations

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Compatibility of HPT with algebraic struc- ture 13

Rooted planar trees

Master equation and HPT

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Higher homotopies, homological pertur- bations, and the working mathemati- cian 25

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