2004/10/29 by Alastair Hamilton, Andrey Lazarev, Hamilton, Alastair +1 · 2 citations
Mathematics · #13D03 #13D10 #46L87 #55P62 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.AT #math.QA #msc:13D03 #msc:13D10 #msc:46L87 #msc:55P62
paper · pdf · doi:10.48550/arxiv.math/0410621
arxiv created 2004/10/29 · openalex publication_date 2004/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study cohomology theories of strongly homotopy algebras, namely A_∞, C_∞ and L_∞-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C_∞-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a C_∞-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C_∞-algebra (an ∞-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincare duality space.