A survey on homological perturbation theory

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30 Pages
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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


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survey on homological perturbation theory
J. Huebschmann
USTL,UFRdeMath´ematiques CNRS-UMR 8524 59655VilleneuvedAscqC´edex,France Johannes.Huebschmann@math.univ-lille1.fr
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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