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An A-infinity operad in spineless cacti
Gálvez Carrillo, Maria Immaculada
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III; Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
The d.g. operad C of cellular chains on the operad of spineless cacti is isomorphic to the Gerstenhaber-Voronov operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad F_2X of the surjection operad. Its homology is the Gerstenhaber operad G. We construct an operad map psi from A-infinity to C such that psi(m_2) is commutative and the homology of psi is the canonical map A \to Com \to G. This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit A-infinty structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map from Com-infinity to C. If such a map could be written down explicitly, it would immediately lead to a G-infinity structure on C and on Hochschild cochains, that is, to a direct proof of the Deligne conjecture
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
Algebraic topology
Homology theory
Categories (Mathematics)
Topologia algebraica
Categories (Matemàtica)
Classificació AMS::18 Category theory; homological algebra::18D Categories with structure
Classificació AMS::55 Algebraic topology::55P Homotopy theory
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