Mediterranean Journal of Mathematics

, Volume 12, Issue 4, pp 1215–1226 | Cite as

An \({\mathcal{A}_\infty}\) Operad in Spineless Cacti

  • Imma Gálvez-Carrillo
  • Leandro Lombardi
  • Andrew Tonks


The dg operad \({\mathcal{C}}\) of cellular chains on the operad of spineless cacti of Kaufmann (Topology 46(1):39–88, 2007) is isomorphic to the Gerstenhaber–Voronov dg operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad \({F_2\mathcal{X}}\) of the surjection operad of Berger and Fresse (Math Proc Camb Philos Soc 137(1):135–174, 2004), McClure and Smith (Recent progress in homotopy theory (Baltimore, MD, 2000). Contemp Math., Amer. Math. Soc., Providence 293:153–193, 2002) and McClure and Smith (J Am Math Soc 16(3):681–704, 2003). Its homology is the Gerstenhaber dg operad \({\mathcal{G}}\). We construct a map of dg operads \({\psi \colon \mathcal{A}_\infty \longrightarrow \mathcal{C}}\) such that \({\psi(m_2)}\) is commutative and \({H_*(\psi)}\) is the canonical map \({\mathcal{A} \to \mathcal{C}\!om \to \mathcal{G}}\). This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit \({\mathcal{A}_\infty}\) structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map \({\mathcal{C}om_\infty \to \mathcal{C}}\). If such a map could be written down explicitly, it would immediately lead to a \({\mathcal{G}_\infty}\) structure on \({\mathcal{C}}\) and on Hochschild cochains, that is, to an explicit and direct proof of the Deligne conjecture.

Mathematics Subject Classification

Primary 18D50 Secondary 55P48 


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© Springer Basel 2015

Authors and Affiliations

  • Imma Gálvez-Carrillo
    • 1
  • Leandro Lombardi
    • 2
  • Andrew Tonks
    • 3
  1. 1.Departament de Matemàtica Aplicada IIIUniversitat Politècnica de CatalunyaTerrassaSpain
  2. 2.Departamento de MatemáticaUniversidad de Buenos Aires, Ciudad UniversitariaBuenos AiresArgentina
  3. 3.Department of MathematicsUniversity of LeicesterLeicesterUK

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