We show that the configuration spaces of a product of parallelizable manifolds may be recovered from those of the factors as the Boardman-Vogt tensor product of right modules over the operads of little cubes of the appropriate dimension. We also discuss an analogue of this result for manifolds that are not necessarily parallelizable, which involves a new operad of skew little cubes. arXiv:1710.05093v2 [math.AT]
We provide spectral Lie algebras with enveloping algebras over the operad of little Gframed n-dimensional disks for any choice of dimension n and structure group G, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincaré-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.
We give explicit formulas for the Betti numbers, both stable and unstable, of
the unordered configuration spaces of an arbitrary surface of finite type.Comment: Minor changes. To appear in the Journal of the London Mathematical
Society. May vary slightly from published versio
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