Origins and breadth of the theory of higher homotopies

TL;DR

This paper reviews the historical development and significance of higher homotopies in mathematics and physics, highlighting their algebraic, topological, and physical applications, and emphasizing the role of homological perturbation theory in understanding them.

Contribution

It provides a comprehensive overview of the origins, evolution, and applications of higher homotopies, connecting classical topology with modern algebraic and physical theories.

Findings

Higher homotopies are fundamental in algebraic topology and physics.

Homological perturbation theory is crucial for practical calculations involving higher homotopies.

Higher homotopies are often implicit and their importance is underappreciated.

Abstract

Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. We recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940's. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor's classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and Ainfty-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950's, has nowadays 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 which, in particular, leads to the effective calculation of various invariants which are otherwise intractable. Higher homotopies 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.