Higher string topology on general spaces

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex X and k ⩾ 1, I construct a spectrum Maps( S k , X ) S ( X ) , which is obtained by taking a generalization of the Spivak bundle on X (which however is not a stable sphere bundle unless X is a Poincaré space), pulling back to Maps( S k , X ) and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the ( k + 1)‐dimensional unframed little disk operad C k + 1 . I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based C k ‐algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad C * C k is Koszul‐dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual C * C k ‐algebras. I show that the cochain complex of X and the chain complex of Ω k X are Koszul dual to each other as C * C k ‐algebras, and that the chain complex of Maps( S k , X ) S ( X ) is naturally equivalent to their (equivalent) Hochschild cohomology in the category of C * C k ‐algebras. 2000 Mathematics Subject Classification 55P48 (primary), 16E40, 55N45, 18D50 (secondary).