In this article, we develop a general technique for gluing subcategories of ∞-categories. We obtain categorical equivalences between simplicial sets associated to certain multisimplicial sets. Such equivalences can be used to construct functors in different contexts. One of our results generalizes Deligne’s gluing theory developed in the construction of the extraordinary pushforward operation i...

This paper is concerned with the theory of cup-products in Hopf-type cyclic cohomology of algebras and coalgebras. Here we give detailed proofs of the statements, announced in our previous paper [21]. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving noncommutative Weil algebra. Then we use a generalization of Quillen and Crainic’s construction (see...

We construct cohomology classes in the space of knots by considering a bundle over this space and “integrating along the fiber” classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes [7], who integrated differential forms along the fiber to get knot invariants. By doing this “in...

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomo...