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An $A_\infty$-version of the Eilenberg-Moore theorem

Matthias Franz

TL;DR

The paper constructs an $A_\infty$-structure on the two-sided bar construction $\mathbf{B}(C^{*}(X),C^{*}(B),C^{*}(E))$ using a homotopy Gerstenhaber algebra (hga) on cochains, extending previous non-associative and dga results. It proves that the induced map to $C^{*}(\tilde{E})$ is a quasi-isomorphism of $A_\infty$-algebras, thereby promoting the Eilenberg–Moore isomorphism to an $A_\infty$-quasi-isomorphism; this yields a canonical product on differential torsion products that agrees with the EM–Smith construction for all triples of spaces. A key technical achievement is showing that the strongly homotopy commutative structure on cochains built by Gugenheim–Munkholm agrees with the one coming from the hga structure, which in turn explains the compatibility of the two bar-product constructions. The results establish naturality with respect to maps of hga triples and provide an explicit description of the higher $m_n$ and $f_n$ in terms of hga operations, linking EM theory to $A_\infty$-transfers and shc structures. This deepens the connection between bar/construction techniques, homotopy-algebraic structures, and classical cohomological computations in fibrations.

Abstract

We construct an $A_\infty$-structure on the two-sided bar construction involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative product defined by Carlson and the author and generalizes the dga structure on the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is promoted to a quasi-isomorphism of $A_\infty$-algebras. We also show that the resulting product on the differential torsion product involving cochain algebras agrees with the one defined by Eilenberg-Moore and Smith, for all triples of spaces. This is a consequence of the following result, which is of independent interest: The strongly homotopy commutative (shc) structure on cochains inductively constructed by Gugenheim-Munkholm agrees with the one previously defined by the author for all hgas.

An $A_\infty$-version of the Eilenberg-Moore theorem

TL;DR

The paper constructs an -structure on the two-sided bar construction using a homotopy Gerstenhaber algebra (hga) on cochains, extending previous non-associative and dga results. It proves that the induced map to is a quasi-isomorphism of -algebras, thereby promoting the Eilenberg–Moore isomorphism to an -quasi-isomorphism; this yields a canonical product on differential torsion products that agrees with the EM–Smith construction for all triples of spaces. A key technical achievement is showing that the strongly homotopy commutative structure on cochains built by Gugenheim–Munkholm agrees with the one coming from the hga structure, which in turn explains the compatibility of the two bar-product constructions. The results establish naturality with respect to maps of hga triples and provide an explicit description of the higher and in terms of hga operations, linking EM theory to -transfers and shc structures. This deepens the connection between bar/construction techniques, homotopy-algebraic structures, and classical cohomological computations in fibrations.

Abstract

We construct an -structure on the two-sided bar construction involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative product defined by Carlson and the author and generalizes the dga structure on the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is promoted to a quasi-isomorphism of -algebras. We also show that the resulting product on the differential torsion product involving cochain algebras agrees with the one defined by Eilenberg-Moore and Smith, for all triples of spaces. This is a consequence of the following result, which is of independent interest: The strongly homotopy commutative (shc) structure on cochains inductively constructed by Gugenheim-Munkholm agrees with the one previously defined by the author for all hgas.
Paper Structure (28 sections, 17 theorems, 178 equations)

This paper contains 28 sections, 17 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities
  4. Sign convention
  5. Bar constructions
  6. The usual bar construction
  7. One- and two-sided bar constructions
  8. Differential torsion products
  9. Homotopy Gerstenhaber algebras
  10. Definition
  11. The product on the right bar construction
  12. A∞-algebras and morphisms
  13. The A∞-structure on the two-sided bar construction
  14. Definition of maps
  15. Some lemmas
  16. ...and 13 more sections

Key Result

Theorem 1.1

The Carlson--Franz product extends to an $A_{\infty}$-structure on the two-sided bar construction $\mathop{\mathrm{\mathbf{B}}}\nolimits\bigl(C^{*}(X),C^{*}(B),C^{*}(E)\bigr)$, and the map eq:intro:def-f1 to a morphism of $A_{\infty}$-algebras (where the target is a dga). Hence $f$ is a quasi-isomorphism of $A_{\infty}$-algebras whenever the map eq:intro:def-f1 is a quasi-isomorphism of complexes

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 25 more