CoHochschild homology of chain coalgebras

Generalizing the work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra C is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C_∗K is its normalized chain complex, then is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on when K is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g,h:K→L, where K and L are reduced, the homology of the coHochschild complex of C_∗L with coefficients in C_∗K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of g and h, and this isomorphism respects comultiplicative structure. In particular, there is an isomorphism, respecting comultiplicative structure, from the homology of to H_∗L|K|, the homology of the free loops on the geometric realization of K.     

Hochschild cohomology of a strongly homotopy commutative algebra

The Hochschild cohomology of a DG algebra A with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. The purpose of this paper is to prove that a certain DG Lie algebra of derivations appears as a finite codimensional graded sub Lie algebra of this Lie algebra when A is a strongly homotopy commutative algebra whose homology is concentrated in finitely many degrees. This result has interesting implications for the free the loop space homology which we explore here as well.     

On a generalised Connes-Hochschild-Kostant-Rosenberg theorem

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character.     

Homology of algebras of families of pseudodifferential operators

We compute the Hochschild, cyclic, and periodic cyclic homology groups of algebras of families of Laurent complete symbols on manifolds with corners. We show in particular that the spectral sequence associated with Hochschild homology degenerates at E² and converges to Hochschild homology. As a byproduct, we identify the space of residue traces on fibrations by manifolds with corners. In the process, we prove some structural results about algebras of complete symbols on manifolds with corners.     

Algebraic geometry of topological spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C ^*-algebras, and for a homology theory of commutative algebras to vanish on C ^*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C ^*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.     

The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character

We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X→Y, the Hochschild complex HX/Y, as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. (2007), doi: 10.1016/j.aim.2007.06.012], decomposes naturally in the derived category D(X) into _⊕p?0Sp(LX/Y[1]), the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case.Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah-Chern character, the exponential of the universal Atiyah class.We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah-Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.     

Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

This article is the sequel to (Marcolli and Tabuada in Sel Math 20(1):315–358, 2014). We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub’s weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin–Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub’s weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky’s category of mixed motives.     

Completeness of the Space of Separable Measures in the Kantorovich-Rubinshtein Metric

We consider the space M(X) of separable measures on the Borel σ-algebra ?(X) of a metric space X. The space M(X) is furnished with the Kantorovich-Rubinshtein metric known also as the “Hutchinson distance” (see [1]). We prove that M(X) is complete if and only if X is complete. We consider applications of this theorem in the theory of selfsimilar fractals.     

When Does the Rational Torsion Split Off for Finitely Generated Modules

It is well known that the torsion part of any finitely generated module over the formal power series ring K[[X]] is a direct summand. In fact, K[[X]] is an algebra dual to the divided power coalgebra over K and the torsion part of any K[[X]]-module actually identifies with the rational part of that module. More generally, for a certain general enough class of coalgebras—those having only finite dimensional subcomodules—we see that the above phenomenon is preserved: the set of torsion elements of any C ^*-module is exactly the rational submodule. With this starting point in mind, given a coalgebra C we investigate when the rational submodule of any finitely generated left C ^*-module is a direct summand. We prove various properties of coalgebras C having this splitting property. Just like in the K[[X]] case, we see that standard examples of coalgebras with this property are the chain coalgebras which are coalgebras whose lattice of left (or equivalently, right, two-sided) coideals form a chain. We give some representation theoretic characterizations of chain coalgebras, which turn out to make a left-right symmetric concept. In fact, in the main result of this paper we characterize the colocal coalgebras where this splitting property holds non-trivially (i.e. infinite dimensional coalgebras) as being exactly the chain coalgebras. This characterizes the cocommutative coalgebras of this kind. Furthermore, we give characterizations of chain coalgebras in particular cases and construct various and general classes of examples of coalgebras with this splitting property.     

A homotopy theoretic realization of string topology

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H _* (LM) of degree -d. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra structure on the S¹ equivariant homology H _* ^{S 1} (LM). These algebraic structures, as well as others, came under the general heading of the ”string topology” of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the ” cactus operad” defined in [6] which is equivalent to operad of framed disks in . This operad action realizes the Chas - Sullivan BV structure on H _* (LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH ^* (C ^* (M); C ^* (M)). Received: 31 July 2001 / Revised version: 11 September 2001 Published online: 5 September 2002     

Embeddings and frames of function spaces

For every Tychonoff space X we denote by Cp(X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of RX. A set P is a frame for the space Cp(X) if Cp(X)⊂P⊂RX. We prove that if Cp(X) embeds in a σ-compact space of countable tightness then X is countable. This shows that it is natural to study when Cp(X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space Cp(X) has a Lindelöf frame of countable tightness then t(X)?ω. We give some generalizations of this result for the case of frames as well as for embeddings of Cp(X) in arbitrary spaces.     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

Normality of products of monotonically normal spaces with compact spaces

Let S be the class of all spaces, each of which is homeomorphic to a stationary subset of a regular uncountable cardinal (depending on the space). In this paper, we prove the following result: The product X×C of a monotonically normal space X and a compact space C is normal if and only if S×C is normal for each closed subspace S in X belonging to S. As a corollary, we obtain the following result: If the product of a monotonically normal space and a compact space is orthocompact, then it is normal.     

Singular homology groups of fuzzy topological spaces

Let L be a completely distributive lattice with order reversing involution, and (X, τ) an L-fuzzy topological space. The purpose of this paper is to introduce the fundamental concept of fuzzy algebraic topology-the singular homology groups of the L-fuzzy topological space, in such a way that they take the (usual) cubical singular homology groups of a topological space as a special case. Also, we shall prove that they are L-fuzzy homeomorphic invariants.     

Finiteness Conditions for Hochschild Homology Algebra and Free Loop Space Cohomology Algebra

We prove that over a characteristic zero field, in most cases, neither the Hochschild homology algebra of a commutative algebra, nor the free loop space cohomology algebra of a topological space, is finitely generated.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Sur l'itération de la construction Cobar

Since Serre's work [12], we know that loop spaces play a central role in algebraic topology. In particular, iterated loop spaces pose the tough problem which consists in iterating the Cobar construction (see [2], [10], [11], [14] and [15]). In this Note, to make progress in this topic, we give explanations and complements about Adams ' relation [1] between C_*(ΩX) and Cobar C_*(X) (Z, Z) when X is the suspension of a reduced (with trivial 0-skeleton) simplicial set, Ω being here the simplicial Kan model [8] of the loop space functor. In particular, our results give a surprising experimental fact: the existence of an “exotic” differential which can replace the classical Adams differential in the Cobar construction. They also permit us to obtain with a new method some previous results of Baues ([2], [3]) about Ω² X when X is the suspension of a 1-reduced (with trivial 1-skeleton) simplicial set.     

Points with maximal Birkhoff average oscillation

Let f: X → X be a continuous map with the specification property on a compact metric space X. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally maximal hyperbolic set.