A survey on homological perturbation theory
by
Johannes Huebschmann
Universit ́ des Sciences et Technologies de Lille Universite des Sciences et Technologies de Lille

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, has become a standard tool to handle algebraic incarnations of higher homotopies. A basic observation is that higher homotopy structures behave much better relative to homotopy than strict structures, and HPT enables one to exploit this observation in various concrete situations. In particular, this leads to the effective calculation of various invariants which are otherwise intractable. The formulas which result from HPT-constructions are recursive, and the calculation of a specific object, e. g. a certain group cohomology group or Lie algebra cohomology group can be carried out, at least in principle, in finitely many steps. Experts on computational algebraic topology (e.g. F. Sergeraert and his collaborators) have already successfully carried out such calculations.

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 foliations.

A basic tool in HPT is the perturbation lemma. Starting from a chain deformation retraction, after introduction of a perturbation of the differential on the big object, this lemma yields recursive formulas for the constituents of a new chain deformation retraction from the perturbed big object onto a perturbed small object. In group cohomology, the perturbation could, for example, correspond to perturbing an abelian group structure to a non-abelian one. At each recursive step, the resulting terms can be determined by explicit programmable algorithms. Sometimes explicit numerical calculations can then be carrried out on the small object. Another basic tool is a lemma providing, under suitable circumstances, a recursive construction for a twisting cochain or, equivalently, solution of the master equation, deformation equation, or Maurer-Cartan equation. Again at each recursive step, the resulting term can be determined by an explicit programmable algorithm.

The talk will illustrate, with a special emphasis on the compatibility of perturbations with algebraic structure and on effective calculation, how HPT may be successfully applied to various mathematical problems arising in group cohomology, algebraic K-theory, deformation theory, differential geometry, physics, etc.

More information about HPT can be found in my survey article arXiv:0710.2645 (in: Gerstenhaber-Stasheff Festschrift, Birkhäuser, to appear) and on my home page.

Paper reference: arXiv:0710.2645

Date received: May 26, 2008