We construct a model of differential K-theory, using the geometrically defined Chern forms, whose cocycles are certain equivalence classes of maps into the Grassmannians and unitary groups. In particular, we produce the circle-integration maps for th
ese models using classical homotopy-theoretic constructions, by incorporating certain differential forms which reconcile the incompatibility between these even and odd Chern forms. By the uniqueness theorem of Bunke and Schick, this model agrees with the spectrum-based models in the literature whose abstract Chern cocycles are compatible with the delooping maps on the nose.
There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is
functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.
We use factorization homology and higher Hochschild (co)chains to study various problems in algebraic topology and homotopical algebra, notably brane topology, centralizers of $E_n$-algebras maps and iterated bar constructions. In particular, we obta
in an $E_{n+1}$-algebra model on the shifted integral chains of the mapping space of the n-sphere into an orientable closed manifold $M$. We construct and use $E_infty$-Poincare duality to identify higher Hochschild cochains, modeled over the $n$-sphere, with the chains on the above mapping space, and then relate Hochschild cochains to the deformation complex of the $E_infty$-algebra $C^*(M)$, thought of as an $E_n$-algebra. We invoke (and prove) the higher Deligne conjecture to furnish $E_n$-Hochschild cohomology, and all that is naturally equivalent to it, with an $E_{n+1}$-algebra structure. We prove that this construction recovers the sphere product. In fact, our approach to the Deligne conjecture is based on an explicit description of the $E_n$-centralizers of a map of $E_infty$-algebras $f:Ato B$ by relating it to the algebraic structure on Hochschild cochains modeled over spheres, which is of independent interest and explicit. More generally, we give a factorization algebra model/description of the centralizer of any $E_n$-algebra map and a solution of Deligne conjecture. We also apply similar ideas to the iterated bar construction. We obtain factorization algebra models for (iterated) bar construction of augmented $E_m$-algebras together with their $E_n$-coalgebras and $E_{m-n}$-algebra structures, and discuss some of its features. For $E_infty$-algebras we obtain a higher Hochschild chain model, which is an $E_n$-coalgebra. In particular, considering an n-connected topological space $Y$, we obtain a higher Hochschild cochain model of the natural $E_n$-algebra structure of the chains of the iterated loop space of $Y$.
In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a
bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [SS]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.
Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential Batalin-Vilkovisky algebra. Moreover, if A is an A-infinity algebra, then TA is a commutative BV-infinity algebra.
This paper generalizes Bismuts equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into
the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe along any closed oriented surface, as well as the derivative of 2-holonomy, which in the case of a torus fits into a sequence of higher holonomies and their differentials.
We show that the tensor product of two cyclic $A_infty$-algebras is, in general, not a cyclic $A_infty$-algebra, but an $A_infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra,
which are contractible polytopes controlling the combinatorial structure of an $A_infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_infty$-algebra can be thought of as an $A_infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $A_infty$-algebras are not necessarily trivial.
In this paper, we study the higher Hochschild functor and its relationship with factorization algebras and topological chiral homology. To this end, we emphasize that the higher Hochschild complex is a $(infty,1)$-functor from the category $hsset tim
es hcdga$ to the category $hcdga$ (where $hsset$ and $hcdga$ are the $(infty,1)$-categories of simplicial sets and commutative differential graded algebras) and give an axiomatic characterization of this functor. From the axioms we deduce several properties and computational tools for this functor. We study the relationship between the higher Hochschild functor and factorization algebras by showing that, in good cases, the Hochschild functor determines a constant commutative factorization algebra. Conversely, every constant commutative factorization algebra is naturally equivalent to a Hochschild chain factorization algebra. Similarly, we study the relationship between the above concepts and topological chiral homology. In particular, we show that on their common domains of definition, the higher Hochschild functor is naturally equivalent to topological chiral homology. Finally, we prove that topological chiral homology determines a locally constant factorization algebra and, further, that this functor induces an equivalence between locally constant factorization algebras on a manifold and (local system of) $E_n$-algebras. We also deduce that Hochschild chains and topological chiral homology satisfies an exponential law, i.e., a Fubini type Theorem to compute them on products of manifolds.
We show that a square zero, degree one element in W(V), the Weyl algebra on a vector space V, is equivalent to providing V with the structure of an algebra over the properad Cobar(coFrob), the properad arising from the cobar construction applied to the cofrobenius coproperad.
We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle.
We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups.
