Coarse homotopy groups

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex. To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts.     

Cohomology of Noncommutative Hilbert Schemes

Noncommutative Hilbert schemes, introduced by M. V. Nori, parametrize left ideals of finite codimension in free algebras. More generally, parameter spaces of finite-codimensional submodules of free modules over free algebras are considered. Cell decompositions of these varieties are constructed, whose cells are parametrized by certain types of forests. Asymptotics for the corresponding Poincaré polynomials and properties of their generating functions are discussed. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary: 16G20; secondary: 14D20.     

Directions for the Long Exact Cohomology Sequence in Moore Categories

A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced by Bourn (Cahiers Topologie Géom Différentielle Catég XL:297–316, 1999; J Pure Appl Algebra 168:133–146, 2002). The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore categories we show that they represent a good context for non-abelian cohomology, in particular for the Baer Extension Theory.      

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type and with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type with coefficients over the module Here the first standard generators of the group act by -multiplication, while the last one acts by -multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type as well as the cohomology of the classical braid group with coefficients in the -dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

     

Loop and Suspension Functors for Small Categories and Stable Homotopy Groups

We investigate a family of -suspension and -loop functors in the category of small categories and relate these families of functors to the classical suspension and loop functors of spaces. We prove also an analogue of the Freudenthal suspension theorem for categories with certain cofibration condition.     

Cohomology theories in triangulated categories

Let C be a triangulated category with a proper class E of triangles.We prove that there exists an Avramov–Martsinkovsky type exact sequence in C,which connects E-cohomology,E-Tate cohomology and E-Gorenstein cohomology.     

On the Derived Invariance of Cohomology Theories for Coalgebras

We study the derived invariance of the cohomology theories Hoch ^*, H ^* and HC ^* associated with coalgebras over a field. We prove a theorem characterizing derived equivalences. As particular cases, it describes the two following situations: (1) f: CD a quasi-isomorphism of differential graded coalgebras, (2) the existence of a cotilting bicomodule _C T _D . In these two cases we construct a derived-Morita equivalence context, and consequently we obtain isomorphisms Hoch ^*(C)Hoch ^*(D) and H ^*(C)H ^*(D). Moreover, when we have a coassociative map inducing an isomorphism H ^*(C)H ^*(D) (for example, when there is a quasi-isomorphism f: CD), we prove that HC ^*(C)HC ^*(D).     

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.     

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

In this paper we introduce and study a cohomology theory {H ⁿ (–,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)} _n0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n3, the functor K(–,n) is right adjoint to the functor _n , where _n (X ) is defined as the fundamental groupoid of the n-loop complex ⁿ (X ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with _i (Y)=0 for all in,n+1 and n3; and also we obtain a classification theorem for those spaces: [–,Y]H ⁿ (–, _n (Y)).     

A fibration of categories overH B

K. A. Hardie and K. H. Kamps investigated the track homotopy categoryH _B over a fixed spaceB ([5]). This paper extendsH _B to the track homotopy categoryH _b over a fixed mapb: , such that there exists a split fibration of categoriesL:H _b →H _B andH _b possesses some construction as inH _B. Supported by National Natural Science Foundation of China and the Doctoral Foundation of the National Educational Committee of China     

Local Projective Model Structures on Simplicial Presheaves

We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Gro-then-dieck site T and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis,A1) and SH((Sch/k)cdh,A1) are equivalent. This result was first proven by Voevodsky and our proof uses many of his techniques, but it does not use his theory of -closed classes.     

Some pairs of adjoint functors involving track homotopy categories

K. A. Hardie and K. H. Kamps investigated the track homotopy categoryH _B over a fixed spaceB ([1]). They have introduced two pairs of adjoint functors:P _B N _B andm _* m ^*, whereP _B :H _B →H ^B , andm _*:H _A →H _B for a fixed mapm:A→B. We have introduced a split fibration of categoriesL:H _b →H _B and provedL J, J L in [2]. This paper first extendsP _B N _B to for any fixed mapb: . Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categoriesH _b andH ^b whereH ^b is the dual ofH _b . One of our results isN _b P _b . This differs fromP _B N _B . Supported by National Natural Science Foundation of China     

On the Action of the Dual Group on the Cohomology of Perverse Sheaves on the Affine Grassmannian

It was proved by Ginzburg, Mirkovic and Vilonen that the G(O)-equivariant perverse sheaves on the affine Grassmannian of a connected reductive group G form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group G . In this paper we construct explicitly the action of G on the global cohomology of a perverse sheaf.     

On the Cohomology of Categories, Universal Toda Brackets and Homotopy Pairs

A category of homotopy pairs is characterised by a cohomology class which generalizes the notion of Toda bracket. Explicit computations of such cohomology classes are described.     

Simplicial Homotopical Algebra and Satellites

We study here a notion of simplicial satellites, as a first step towards a characterisation of simplicial derived functors, a problem unsolved since the latter were introduced.The problem comes from the fact that, in contrast with the abelian case, simplicial derived functors do not produce by themselves an exact sequence. Our solution consists in extending them to commutative k-cubes, for all k, forming thus an exact system of functors universal within the connected ones; or, in other words, a system of simplicial satellites. The tool we develop here for this extension is the homotopy kernel of a commutative k-dimensional cubic diagram, generalising the homotopy kernel of a map; its 2-dimensional version has already been proved essential in other homotopical topics.     

Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

Cohomology of the Universal Enveloping Algebras of Certain Bigraded Lie Algebras

Let p be an odd prime and q = 2(p-1).Up to total degree t-s max{(5p~3+ 6p~2+ 6 p +4)q-10,p~4q},the generators of H~(s,t)(U(L)),the cohomology of the universal enveloping algebra of a bigraded Lie algebra L,are determined and their convergence is also verified.Furthermore our results reveal that this cohomology satisfies an analogous Poinare duality property.This largely generalizes an earlier classical results due to J.P.May.     

Homotopy coherent category theory

This article is an introduction to the categorical theory of homotopy coherence. It is based on the construction of the homotopy coherent analogues of end and coend, extending ideas of Meyer and others. The paper aims to develop homotopy coherent analogues of many of the results of elementary category theory, in particular it handles a homotopy coherent form of the Yoneda lemma and of Kan extensions. This latter area is linked with the theory of generalised derived functors.

     

Cohomology of classical algebraic groups from the functorial viewpoint

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.