Volume 17, Issue 1

We construct an example of an 𝐴 ∞ algebra structure defined over a finite dimensional graded vector space.

Let 𝐺 × τ 𝐺′ be the principal twisted Cartesian product with fibre 𝐺, base 𝐺 and twisting function where 𝐺 and 𝐺′ are simplicial groups as well as 𝐺 × τ 𝐺′; and 𝐶 𝑁 (𝐺) ⊗ 𝑡 𝐶 𝑁 (𝐺′) be the twisted tensor product associated to 𝐶 𝑁 (𝐺 × τ 𝐺′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair 𝐶 𝑁 (𝐺) ⊗ 𝑡 𝐶 𝑁 (𝐺′), μ ) is a multiplicative Cartan's construction where μ is the standard product on 𝐶 𝑁 (𝐺) ⊗ 𝐶 𝑁 (𝐺′). Furthermore, assuming that a contraction from 𝐶 𝑁 (𝐺′) to 𝐻𝐺′ exists and using the techniques from homological perturbation theory, we extend the former result to other “twisted” tensor products of the form 𝐶 𝑁 (𝐺) ⊗ 𝐻𝐺′.

We describe a conjecture on the algebra of higher cohomology operations which leads to the computations of the differentials in the Adams spectral sequence. For this we introduce the notion of an 𝑛-th order track category suitable for studying higher order Toda brackets and the differentials in spectral sequences. We describe various examples of higher order track categories which are topological, in particular the track category of higher cohomology operations. Also, differential algebras give rise to higher order track categories.

The computer program Kenzo is used to study complex 𝐴 ∞ -structures coming from iterated loop spaces. The methods of constructive algebraic topology , due to the authors, do produce chain complexes of finite type and chain equivalences with the Hopf algebras canonically associated to the loop spaces. These chain complexes of finite type are therefore endowed with 𝐴 ∞ -structures. It is then experimentally observed, using the Kenzo program, that 𝐴 ∞ -structures of arbitrarily high orders are nontrivial in these objects. This case study is a good illustration of the computing tools given to the topologists by the Kenzo program.

We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴 ∞ -algebras.

We relate a construction of Kadeishvili's establishing an 𝐴 ∞ -structure on the homology of a differential graded algebra or, more generally, of an 𝐴 ∞ -algebra with certain constructions of Chen and Gugenheim. Thereafter we establish the links of these constructions with subsequent developments.

Early in the history of higher homotopy algebra [Stasheff, Trans. Am. Math. Soc. 108: 293–312, 1963], it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeishvili that showed they were but a shadow of an 𝐴 ∞ -structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim [J. Pure Appl. Algebra 25: 197–205, 1982] and K. T. (Chester) Chen [Ann. of Math. (2) 97: 217–246, 1973]. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.