  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 $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, and show that
the corresponding chain complex is naturally homotopy equivalent to an algebra
over the $(k+1)$-dimensional unframed little disk operad $\mathcal{C}_{k+1}$. I
also prove Kontsevich's conjecture that the Quillen cohomology of a based
$\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a
shift of its Hochschild cohomology, as well as prove that the operad
$C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived
category. This gives one a natural notion of (derived) Koszul dual
$C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and
the chain complex of $\Omega^k X$ are Koszul dual to each other as
$C_{\ast}\mathcal{C}_k$-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 $C_{\ast}\mathcal{C}_k$-algebras.
