Homology, Homotopy and Applications Articles (Project Euclid)

A universal property for $Sp(2)$ at the prime $3$

Thu, 05 Aug 2010 15:41 EDT

Jelena Grbić, Stephen Theriault

Source: Homology Homotopy Appl., Volume 11, Number 1, 1--15.

Abstract:
We study a universal property of $Sp(2)$ in the category of $3$-local homotopy associative, homotopy commutative $H$-spaces. We show that while $Sp(2)$ fails to be universal in the full category, there is a subcategory in which it is universal for its $7$-skeleton.

Colocalization functors in derived categories and torsion theories

Fri, 29 Jul 2011 11:29 EDT

Shoham Shamir

Source: Homology Homotopy Appl., Volume 13, Number 1, 75--88.

Abstract:
Let $R$ be a ring and let $\mathcal{A}$ be a hereditary torsion class of $R$-modules. The inclusion of the localizing subcategory generated by $\mathcal{A}$ into the derived category of $R$ has a right adjoint, denoted CellA. Recently, Benson has shown how to compute $\operatorname{Cell}_{\mathcal{A}}R$ when $R$ is a group ring of a finite group over a prime field and $\mathcal{A}$ is the hereditary torsion class generated by a simple module. We generalize Benson’s construction to the case where $\mathcal{A}$ is any hereditary torsion class on $R$. It is shown that for every $R$-module $M$ there exists an injective $R$-module $E$ such that: $$H^n(\operatorname{Cell}_{\mathcal{A}}M)\cong \operatorname{Ext}^{n-1}_{\operatorname{End}_R(E)} (\operatorname{Hom}_R (M,E),E)\hbox{ for }n\ge 2. $$

On the 3-arrow calculus for homotopy categories

Fri, 29 Jul 2011 11:29 EDT

Sebastian Thomas

Source: Homology Homotopy Appl., Volume 13, Number 1, 89--119.

Abstract:
We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. Applications include the localisation of an arbitrary Quillen model category with respect to its weak equivalences as well as the localisation of its full subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy category in all four cases. In contrast to the approach of Dwyer, Hirschhorn, Kan and Smith, the Quillen model category under consideration does not need to admit functorial factorisations.

The Brown-Golasiński model structure on strict ∞-groupoids revisited

Fri, 29 Jul 2011 11:29 EDT

Dimitri Ara, François Métayer

Source: Homology Homotopy Appl., Volume 13, Number 1, 121--142.

Abstract:
We prove that the folk model structure on strict $\infty$-categories transfers to the category of strict $\infty$-groupoids (and more generally to the category of strict $(\infty; n)$-categories), and that the resulting model structure on strict $\infty$-groupoids coincides with the one defined by Brown and Golasiński via crossed complexes.

Smooth functors vs. differential forms

Fri, 29 Jul 2011 11:29 EDT

Urs Schreiber, Konrad Waldorf

Source: Homology Homotopy Appl., Volume 13, Number 1, 143--203.

Abstract:
We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as derivatives of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.

Generalized Davis-Januszkiewicz spaces, multicomplexes and monomial rings

Fri, 29 Jul 2011 11:29 EDT

Alvise J. Trevisan

Source: Homology Homotopy Appl., Volume 13, Number 1, 205--221.

Abstract:
We show that every monomial ring can be realized topologically by a certain topological space. This space is called a generalized Davis-Januszkiewicz space and can be thought of as a colimit over a multicomplex, a combinatorial object generalizing a simplicial complex. Furthermore, we show that such a space is obtained as the homotopy fiber of a certain map with total space the classical Davis-Januszkiewicz space.

p-local finite group cohomology

Fri, 29 Jul 2011 11:29 EDT

Ran Levi, Kári Ragnarsson

Source: Homology Homotopy Appl., Volume 13, Number 1, 223--257.

Abstract:
We study cohomology for $p$-local finite groups with nonconstant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences.

Operations on the Hopf-Hochschild complex for module-algebras

Fri, 29 Jul 2011 11:29 EDT

Donald Yau

Source: Homology Homotopy Appl., Volume 13, Number 1, 259--272.

Abstract:
It is proved that Kaygun’s Hopf-Hochschild cochain complex for a module-algebra is a brace algebra with multiplication. As a result, an analogue of Deligne’s Conjecture holds for module-algebras, and the Hopf-Hochschild cohomology of a module-algebra has a Gerstenhaber algebra structure.

Steenrod's operations in simplicial Bredon-Illman cohomology with local coefficients

Fri, 29 Jul 2011 11:29 EDT

Goutam Mukherjee, Debasis Sen

Source: Homology Homotopy Appl., Volume 13, Number 1, 273--296.

Abstract:
In this paper we use Peter May’s algebraic approach to Steenrod operations to construct Steenrod’s reduced power operations in simplicial Bredon-Illman cohomology with local coefficients of a one vertex $G$-Kan complex, $G$ being a discrete group.

Cyclic structures in algebraic (co)homology theories

Fri, 29 Jul 2011 11:29 EDT

Niels Kowalzig, Ulrich Krähmer

Source: Homology Homotopy Appl., Volume 13, Number 1, 297--318.

Abstract:
This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel’d modules

Cech approximation to the Brown-Gersten spectral sequence

Fri, 29 Jul 2011 11:29 EDT

Benjamin Antieau

Source: Homology Homotopy Appl., Volume 13, Number 1, 319--348.

Abstract:
In this paper, we show that the étale index of a torsion cohomological Brauer class is divisible by the period of the class. The tool used to make this computation is the Čech approximation of the title. To create the approximation, we use the folklore theorem that the homotopy limit and Postnikov spectral sequences for a cosimplicial space agree beginning with the $E_2$-page. As far we know, this folklore theorem has no proof in the literature, so we include a proof.

Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space

Fri, 29 Jul 2011 11:29 EDT

Katsuhiko Kuribayashi

Source: Homology Homotopy Appl., Volume 13, Number 1, 349--379.

Abstract:
Let $M$ be a homogeneous space admitting a left translation by a connected Lie group $G$. The adjoint to the action gives rise to a map from $G$ to the monoid of self-homotopy equivalences of $M$. The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy group. In particular, the visibility degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik.

Productive elements in group cohomology

Fri, 29 Jul 2011 11:29 EDT

Ergün Yalçin

Source: Homology Homotopy Appl., Volume 13, Number 1, 381--401.

Abstract:
Let $G$ be a finite group and $k$ be a field of characteristic $p > 0$. A cohomology class $\zeta\in H^n (G,k)$ is called productive if it annihilates $\operatorname{Ext}^*_{kG}(L_\zeta,L_\zeta)$. We consider the chain complex $\mathbf{P}(\zeta)$ of projective $kG$-modules which has the homology of an $(n - 1)$-sphere and whose $k$-invariant is $\zeta$ under a certain polarization. We show that $\zeta$ is productive if and only if there is a chain map $\Delta : \mathbf{P}(\zeta)\to \mathbf{P}(\zeta)\otimes \mathbf{P}(\zeta)$ such that $(\operatorname{id} \otimes \epsilon) \Delta \simeq \operatorname{id}$ and $(\epsilon \otimes \operatorname{id}) \Delta \simeq \operatorname{id}$. Using the Postnikov decomposition of $\mathbf{P}(\zeta) \otimes \mathbf{P}(\zeta)$, we prove that there is a unique obstruction for constructing a chain map $\Delta$ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements.

Erratum to 'Dicovering spaces'

Fri, 29 Jul 2011 11:29 EDT

Lisbeth Fajstrup

Source: Homology Homotopy Appl., Volume 13, Number 1, 403--406.

Abstract:
In [ 1 ] (L. Fajstrup, Dicovering spaces, Homology Homotopy Appl. 5 (2003), no. 2, 1–17), we study coverings in the setting of directed topology. Unfortunately, there is a condition missing in the definition of a directed covering. Some of the results in [ 1 ] require this extra condition and in fact it was claimed to follow from the original definition. It is the purpose of this note to give the right definition and point out how this affects the statements in that paper. Moreover, we give an example of a dicovering in the sense of [ 1 ], which does not satisfy the extra condition. Fortunately, with the extra condition, the subsequent results are now correct.

Lifting of model structures to fibred categories

Mon, 30 Apr 2012 13:25 EDT

Abhishek Banerjee

Source: Homology Homotopy Appl., Volume 13, Number 2, 1--17.

Abstract:
A fibred category consists of a functor $p: \mathbf{N} \to \mathbf{M}$ between categories $\mathbf{N}$ and $\mathbf{M}$ such that objects of $\mathbf{N}$ may be "pulled back along any arrow of $\mathbf{M}$". Given a fibred category $p: \mathbf{N} \to \mathbf{M}$ and a model structure on the "base category" $\mathbf{M}$, we show that there exists a lifting of the model structure on $\mathbf{M}$ to a model structure on $\mathbf{N}$. We will refer to such a system as a "fibred model category" and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base category $\mathbf{M}$ may be lifted to right homotopic maps in the fibred category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf{N}$ and $\mathbf{M}$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf{M}$ lifts to a Quillen pair on $\mathbf{N}$.

On the vanishing of cohomology in triangulated categories

Mon, 30 Apr 2012 13:25 EDT

Petter Andreas Bergh

Source: Homology Homotopy Appl., Volume 13, Number 2, 19--36.

Abstract:
We study the vanishing of cohomology in a triangulated category, in particular vanishing gaps and symmetry.

A generalization of the Wang sequence

Mon, 30 Apr 2012 13:25 EDT

Haibao Duan

Source: Homology Homotopy Appl., Volume 13, Number 2, 37--42.

Abstract:
The classical Wang sequence (for cohomology of fiber bundles over a sphere) is extended to a more generalized setting, given by gluing together two disc bundles over manifolds along their boundaries.

Rational homotopy models for two-point configuration spaces of lens spaces

Mon, 30 Apr 2012 13:25 EDT

Matthew S. Miller

Source: Homology Homotopy Appl., Volume 13, Number 2, 43--62.

Abstract:
We study the algebraic topology of configuration spaces as interesting objects in their own right and with the goal of constructing invariants for topological manifolds. We calculate the complete Massey product structure for the universal cover of the space of two point configurations in a three-dimensional lens space. We then construct rational homotopy models for these spaces and calculate the rational homotopy groups.

On the $K$-theory and homotopy theory of the Klein bottle group

Mon, 30 Apr 2012 13:25 EDT

Jens Harlander, Andrew Misseldine

Source: Homology Homotopy Appl., Volume 13, Number 2, 63--72.

Abstract:
We construct infinitely many chain homotopically distinct algebraic 2-complexes for the Klein bottle group and give various topological applications. We compare our examples to other examples in the literature and address the question of geometric realizability.

Coarse geometry and P. A. Smith theory

Mon, 30 Apr 2012 13:25 EDT

Ian Hambleton, Lucian Savin

Source: Homology Homotopy Appl., Volume 13, Number 2, 73--102.

Abstract:
We define a generalization of the fixed point set, called the bounded fixed set, for a group acting by isometries on a metric space. An analogue of the P. A. Smith theorem is proved for finite p-group actions on metric spaces of finite asymptotic dimension, which relates the coarse homology of the bounded fixed set to the coarse homology of the total space.

On the algebraic $K$-theory of the coordinate axes over the integers

Mon, 30 Apr 2012 13:25 EDT

Vigleik Angeltveit, Teena Gerhardt

Source: Homology Homotopy Appl., Volume 13, Number 2, 103--111.

Abstract:
We show that the relative algebraic $K$-theory group $K_{2i}(\mathbb{Z}[x, y]/(xy), (x, y))$ is free abelian of rank 1 and that $K_{2i+1}(\mathbb{Z}[x, y]/(xy), (x, y))$ is finite of order $(i!)^2$. We also find the group structure of $K_{2i+1}(\mathbb{Z}[x, y]/(xy), (x, y))$ in low degrees.

On Groebner bases and immersions of Grassmann manifolds $G_{2,n}$

Mon, 30 Apr 2012 13:25 EDT

Zoran Z. Petrović, Branislav I. Prvulović

Source: Homology Homotopy Appl., Volume 13, Number 2, 113--128.

Abstract:
Mod 2 cohomology of the Grassmann manifold $G_{2,n}$ is a polynomial algebra modulo a certain well-known ideal. A Groebner basis for this ideal is obtained. Using this basis, some new immersion results for Grassmannians $G_{2,n}$ are established.

The fundamental 2-crossed complex of a reduced CW-complex

Mon, 30 Apr 2012 13:25 EDT

João Faria Martins

Source: Homology Homotopy Appl., Volume 13, Number 2, 129--157.

Abstract:
We define the fundamental 2-crossed complex $\Omega^\infty(X)$ of a reduced CW-complex $X$ from Ellis’ fundamental squared complex $\rho^\infty(X)$ thereby proving that $\Omega^\infty(X)$ is totally free on the set of cells of $X$. This fundamental 2-crossed complex has very good properties with regard to the geometrical realisation of 2-crossed complex morphisms. After carefully discussing the homotopy theory of totally free 2-crossed complexes, we use $\Omega^\infty(X)$ to give a new proof that the homotopy category of pointed 3-types is equivalent to the homotopy category of 2-crossed modules of groups. We obtain very similar results to the ones given by Baues in the similar context of quadratic modules and quadratic chain complexes.

Remark on rigidity over several fields

Mon, 30 Apr 2012 13:25 EDT

Serge Yagunov

Source: Homology Homotopy Appl., Volume 13, Number 2, 159--164.

Abstract:
It is shown that T-spectrum representable cohomology theories on smooth algebraic varieties satisfy normalization condition over nonreal fields. As a consequence, one can see that the rigidity property holds for all representable theories over considered fields.

Topological invariance of the combinatorial Euler characteristic of tame spaces

Mon, 30 Apr 2012 13:25 EDT

Tibor Beke

Source: Homology Homotopy Appl., Volume 13, Number 2, 165--174.

Abstract:
We prove the topological invariance of the combinatorial Euler characteristic with the help of a canonical, topologically defined stratification of tame spaces by locally compact, tame strata.

L-infinity maps and twistings

Mon, 30 Apr 2012 13:25 EDT

Joseph Chuang, Andrey Lazarev

Source: Homology Homotopy Appl., Volume 13, Number 2, 175--195.

Abstract:
We give a construction of an $L_\infty$ map from any $L_\infty$ algebra into its truncated Chevalley-Eilenberg complex as well as its cyclic and $A_\infty$ analogues. This map fits with the inclusion into the full Chevalley-Eilenberg complex (or its respective analogues) to form a homotopy fiber sequence of $L_\infty$ algebras. Applications to deformation theory and graph homology are given. We employ the machinery of Maurer-Cartan functors in $L_\infty$ and $A_\infty$ algebras and associated twistings which should be of independent interest.

Support varieties and representation type of self-injective algebras

Mon, 30 Apr 2012 13:25 EDT

Jörg Feldvoss, Sarah Witherspoon

Source: Homology Homotopy Appl., Volume 13, Number 2, 197--215.

Abstract:
We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in "Support varieties and representation type of small quantum groups," Internat. Math. Res. Notices 2010, no. 7, 1346–1362. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.

Comparing operadic theories of $n$-category

Mon, 30 Apr 2012 13:25 EDT

Eugenia Cheng

Source: Homology Homotopy Appl., Volume 13, Number 2, 217--249.

Abstract:
We give a framework for comparing on the one hand theories of $n$-categories that are weakly enriched operadically, and on the other hand $n$-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by a $\mathcal{V}$-operad $P$. We define weak $n$-categories by iterated weak enrichment using a series of parametrising operads $P_i$. We then show how to construct from such a theory an $n$-dimensional globular operad for each $n \geqslant 0$ whose algebras are precisely the weak $n$-categories, and we show that the resulting globular operad is contractible precisely when the operads $P_i$ are contractible. We then show how the globular operad associated with Trimble’s topological definition is related to the globular operad used by Batanin to define fundamental $n$-groupoids of spaces.

Tensor products of homotopy Gerstenhaber algebras

Mon, 30 Apr 2012 13:25 EDT

Matthias Franz

Source: Homology Homotopy Appl., Volume 13, Number 2, 249--262.

Abstract:
On the tensor product of two homotopy Gerstenhaber algebras we construct a Hirsch algebra structure which extends the canonical dg algebra structure. Our result applies more generally to tensor products of “level 3 Hirsch algebras” and also to the Mayer–Vietoris double complex.

Lie coalgebras and rational homotopy theory, I: graph coalgebras

Mon, 30 Apr 2012 13:25 EDT

Dev Sinha, Benjamin Walter

Source: Homology Homotopy Appl., Volume 13, Number 2, 263--292.

Abstract:
We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.

On the orientability of the slice filtration

Mon, 30 Apr 2012 13:25 EDT

Pablo Pelaez

Source: Homology Homotopy Appl., Volume 13, Number 2, 293--300.

Abstract:
Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\mathcal{SH}$ are strict modules over Voevodsky’s algebraic cobordism spectrum. We also show that the zero slice of any commutative ring spectrum in $\mathcal{SH}$ is an oriented ring spectrum in the sense of Morel, and that its associated formal group law is additive. As a consequence, we deduce that with rational coefficients the slices are in fact motives in the sense of Cisinski-Déglise and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky.

Stability for closed surfaces in a background space

Mon, 30 Apr 2012 13:25 EDT

Ralph L. Cohen, Ib Madsen

Source: Homology Homotopy Appl., Volume 13, Number 2, 301--313.

Abstract:
In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $\mathscr{S}_g(K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $\mathscr{S}_{g,n}(K)$, was studied by the authors in a previous paper. The study there relied on stability results for the homology of mapping class groups, $\Gamma_{g,n}$ with certain families of twisted coefficients. It turns out that these mapping class groups only have homological stability when $n$, the number of boundary components, is positive, or in the closed case when the coefficient modules are trivial. Because of this we present a new proof of the rational homological stability for $\mathscr{S}_g(K)$, that is homotopy theoretic in nature. We also take the opportunity to prove a new stability theorem for closed surfaces in $K$ that have marked points.

Co-representability of the Grothendieck group of endomorphisms functor in the category of noncommutative motives

Mon, 30 Apr 2012 13:25 EDT

Goncalo Tabuada

Source: Homology Homotopy Appl., Volume 13, Number 2, 315--328.

Abstract:
In this article we prove that the additive invariant corepresented by the noncommutative motive $\mathbb{Z}[r]$ is the Grothendieck group of endomorphisms functor $K_0\mathrm{End}$. Making use of Almkvist’s foundational work, we then show that the ring $\mathrm{Nat}(K_0\mathrm{End},K_0\mathrm{End})$ of natural transformations (whose multiplication is given by composition) is naturally isomorphic to the direct sum of $\mathbb{Z}$ with the ring $W_0(\mathbb{Z}[r])$ of fractions of polynomials with coefficients in $\mathbb{Z}[r]$ and constant term 1.

Motives of some acyclic varieties

Mon, 30 Apr 2012 13:25 EDT

Aravind Asok

Source: Homology Homotopy Appl., Volume 13, Number 2, 329--335.

Abstract:
We prove that the Voevodsky motive with $\mathbb{Z}$-coefficients (resp. $\mathbb{Q}$-coefficients) of a $\mathbb{Z}$-acylic (resp. $\mathbb{Q}$-acyclic) smooth complex variety of dimension $\leqslant 2$ is isomorphic to that of a point and discuss some related extensions.

The isomorphism conjecture in $L$-theory: Graphs of groups

Wed, 12 Dec 2012 09:04 EST

S. K. Roushon

Source: Homology Homotopy Appl., Volume 14, Number 1, 1--17.

Abstract:
We study the fibered isomorphism conjecture of Farrell and Jones in $L$-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in 2. of Theorem 1.2, we prove the $L$-theory version of Theorem 1.2 in the 2003 paper by Farrell and Linnell.

A homotopy colimit theorem for diagrams of braided monoidal categorie

Wed, 12 Dec 2012 09:04 EST

A. R. Garzón, R. Pérez

Source: Homology Homotopy Appl., Volume 14, Number 1, 19--32.

Abstract:
Thomason’s Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can also be represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams B of braided monoidal categories, by means of a simplicial set nerve of the diagram. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplicial sets, obtained from composing B with the geometric nerve functor of braided monoidal categories.

Semicoverings: a generalization of covering space theory

Wed, 12 Dec 2012 09:04 EST

Jeremy Brazas

Source: Homology Homotopy Appl., Volume 14, Number 1, 33--63.

Abstract:
Using universal constructions of topological groups, one can endow the fundamental group of a space with a topology and obtain a topological group. Additionally, the fundamental groupoid of a space becomes enriched over Top when the homsets are endowed with similar topologies. This paper is devoted to a generalization of classical covering theory in the context of these constructions.

Jacobi-Zariski exact sequence for Hochschild homology and cyclic (co)homology

Wed, 12 Dec 2012 09:04 EST

Atabey Kaygun

Source: Homology Homotopy Appl., Volume 14, Number 1, 65--78.

Abstract:
We prove that for an inclusion of unital associative but not necessarily commutative $\mathbb{k}$-algebras $\mathcal{B}\subseteq \mathcal{A}$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $\mathcal{B}$-module $\mathcal{A}/\mathcal{B}$ is flat. We also prove that for an arbitrary $r$-flat morphism $\varphi\colon\mathcal{B}\to\mathcal{A}$ with an H-unital kernel, one can express the Wodzicki excision sequence and our Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.

Normal and conormal maps in homotopy theory

Wed, 12 Dec 2012 09:04 EST

Emmanuel D. Farjoun, Kathryn Hess

Source: Homology Homotopy Appl., Volume 14, Number 1, 79--112.

Abstract:
Let $M$ be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in $M$. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normalized chain complex functor. We provide several explicit classes of examples of homotopynormal and of homotopy-conormal maps, when $M$ is the category of simplicial sets or the category of chain complexes over a commutative ring.

Morphic cohomology of toric varieties

Wed, 12 Dec 2012 09:04 EST

Abdó Roig-Maranges

Source: Homology Homotopy Appl., Volume 14, Number 1, 113--132.

Abstract:
In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even in the singular case) in terms of combinatorial data coming from the fan of the toric variety.

Cubical approach to derived functors

Wed, 12 Dec 2012 09:04 EST

Irakli Patchkoria

Source: Homology Homotopy Appl., Volume 14, Number 1, 133--158.

Abstract:
We construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors and prove that these cubical derived functors are naturally isomorphic to their simplicial counterparts. We also show that this result generalizes the well-known fact that the simplicial and cubical singular homologies of a topological space are naturally isomorphic.

Computing braid groups of graphs with applications to robot motion planning

Wed, 12 Dec 2012 09:04 EST

Vitaliy Kurlin

Source: Homology Homotopy Appl., Volume 14, Number 1, 159--180.

Abstract:
An algorithm is designed to write down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given connected graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and is quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.

Scissors congruence as $K$-theory

Wed, 12 Dec 2012 09:04 EST

Inna Zakharevich

Source: Homology Homotopy Appl., Volume 14, Number 1, 181--202.

Abstract:
Scissors congruence groups have traditionally been expressed algebraically in terms of group homology. We give an alternate construction of these groups by producing them as the $0$-level in the algebraic $K$-theory of a Waldhausen category.

Potentials of homotopy cyclic A-infinity algebras

Wed, 12 Dec 2012 09:04 EST

Cheol-Hyun Cho, Sangwook Lee

Source: Homology Homotopy Appl., Volume 14, Number 1, 203--220.

Abstract:
For a cyclic A-infinity algebra, a potential recording the structure constants can be defined. We define an analogous potential for a homotopy cyclic A-infinity algebra and prove its properties. On the other hand, we find another different potential for a homotopy cyclic A-infinity algebra, which is related to the algebraic analogue of generalized holonomy map of Abbaspour, Tradler and Zeinalian.

A fast algorithm for constructing topological structure in large data

Wed, 12 Dec 2012 09:04 EST

Xu Liu, Zheng Xie, Dongyun Yi

Source: Homology Homotopy Appl., Volume 14, Number 1, 221--238.

Abstract:
Discovering and constructing the topological structure in data has attracted the attention within the community of data analysis. However, most methods developed so far are unsuitable for very large sets of data because of their computational difficulties. This paper presents a fast algorithm for constructing the inherent topological structure in large sets of data that might be noisy in order to enhance the MAPPER algorithm introduced by Singh, Mémoli and Carlsson. The limitation of our method, as shown by our experiments, lies with the storage in the main memory rather than the computing time.

Weight structures and 'weights' on the hearts of $t$-structures

Wed, 12 Dec 2012 09:04 EST

Mikhail V. Bondarko

Source: Homology Homotopy Appl., Volume 14, Number 1, 239--261.

Abstract:
We define and study transversal weight and $t$-structures (for triangulated categories); if a weight structure is transversal to a $t$-one, then it defines certain 'weights' for its heart. Our results axiomatize and describe in detail the relations between the Chow weight structure $w_{Chow}$ for Voevodsky's motives (introduced in a preceding paper), the (conjectural) motivic $t$-structure, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of Deligne's 1-motives (over a smooth base) and of Artin-Tate motives over number fields. In particular, we prove that the 'weights' for Voevodsky's motives (that are given by $w_{Chow}$) are compatible with those for 1-motives (that were 'classically' defined using a quite distinct method); this result is new. Weight structures transversal to the canonical $t$-structures also exist for the Beilinson's $D^b_{\bar{H}_p}$ (the derived category of graded polarizable mixed Hodge complexes) and for the derived category of (Saito's) mixed Hodge modules. We also study weight filtrations for the heart of $t$ and (the degeneration of) weight spectral sequences. The corresponding relation between $t$ and $w$ is strictly weaker than transversality; yet it is easier to check, and we still obtain a certain filtration for (objects of) the heart of $t$ that is strictly respected by morphisms. In a succeeding paper we apply the results obtained in order to reduce the existence of Beilinson’s mixed motivic sheaves (over a base scheme $S$) and 'weights' for them to (certain) standard motivic conjectures over a universal domain $K$.

Class-combinatorial model categories

Wed, 12 Dec 2012 09:04 EST

Boris Chorny, Jiří Rosický

Source: Homology Homotopy Appl., Volume 14, Number 1, 263--280.

Abstract:
We develop an extension of the framework of combinatorial model categories. The category of small presheaves over large indexing categories and ind-categories are the basic examples of non-combinatorial model categories embraced by the new machinery called class-combinatorial model categories. The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper, where we introduced the concepts of class-locally presentable and class-accessible categories. In this work we prove that the category of weak equivalences of a nice class-combinatorial model category is class-accessible. Our extension of J. Smith’s localization theorem depends on the verification of a cosolution-set condition. The deepest result is that the (left Bousfield) localization of a class-combinatorial model category with respect to a strongly class-accessible localization functor is class-combinatorial again.

The isomorphism between motivic cohomology and $K$-groups for equi-characteristic regular local rings

Wed, 12 Dec 2012 09:04 EST

Yuki Kato

Source: Homology Homotopy Appl., Volume 14, Number 1, 281--285.

Abstract:
One of the well-known problems in algebraic $K$-theory is the comparison of higher Chow groups and $K$-groups. In this paper, using the motivic complex defined by Voevodsky– Suslin–Friedlander, we prove the comparison theorem for equi-characteristic regular local rings.

Erratum to "Adding inverses to diagrams encoding algebraic structures" and "Adding inverses to diagrams II: Invertible homotopy theories are spaces"

Wed, 12 Dec 2012 09:04 EST

Julia E. Bergner

Source: Homology Homotopy Appl., Volume 14, Number 1, 287--291.

Abstract:
In this note, we correct an error in one of our approaches to encode a group structure by a diagram. We show that we instead obtain the structure of a monoid with involution.

3 X 3 lemma for star-exact sequences

Wed, 12 Dec 2012 09:11 EST

Marino Gran, Zurab Janelidze, Diana Rodelo

Source: Homology Homotopy Appl., Volume 14, Number 2, 1--22.

Abstract:
A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding $3 \times 3$ lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with “exact forks” and then, the corresponding $3 \times 3$ lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such $3 \times 3$ lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.

A representability theorem for some huge abelian categories

Wed, 12 Dec 2012 09:11 EST

George Ciprian Modoi

Source: Homology Homotopy Appl., Volume 14, Number 2, 23--36.

Abstract:
We define quasi-locally presentable categories as big unions of a chain of coreflective subcategories that are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi-locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi-locally presentable, and we obtain a new proof of the Brown representability theorem. Examples of functors that are not representable are also given.

Matrix factorizations over projective schemes

Wed, 12 Dec 2012 09:11 EST

Jesse Burke, Mark E. Walker

Source: Homology Homotopy Appl., Volume 14, Number 2, 37--61.

Abstract:
We study matrix factorizations of regular global sections of line bundles on schemes. If the line bundle is very ample relative to a Noetherian affine scheme we show that morphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we prove an analogue of Orlov’s theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. Moreover, we give a complete description of the image of this functor.

Formality of Koszul brackets and deformations of holomorphic Poisson manifolds

Wed, 12 Dec 2012 09:11 EST

Domenico Fiorenza, Marco Manetti

Source: Homology Homotopy Appl., Volume 14, Number 2, 63--75.

Abstract:
We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrangian submanifold, endowed with the Koszul bracket. As a corollary we generalize a recent result by Hitchin on deformations of holomorphic Poisson manifolds.

Grid diagrams and shellability

Wed, 12 Dec 2012 09:11 EST

Sucharit Sarkar

Source: Homology Homotopy Appl., Volume 14, Number 2, 77--90.

Abstract:
We explore a somewhat unexpected connection between knot Floer homology and shellable posets, via grid diagrams. Given a grid presentation of a knot $K$ inside $S^3$, we define a poset which has an associated chain complex whose homology is the knot Floer homology of $K$. We then prove that the closed intervals of this poset are shellable. This allows us to combinatorially associate a PL flow category to a grid diagram.

Automorphisms of Hurwitz Series

Wed, 12 Dec 2012 09:11 EST

William F. Keigher, Varadharaj R. Srinivasan

Source: Homology Homotopy Appl., Volume 14, Number 2, 91--99.

Abstract:
In this article we will define the notions of Hurwitz automorphism and comorphism of the ring of Hurwitz series. A Hurwitz automorphism is the analog of a Seidenberg automorphism of a power series ring when the characteristic of the underlying ring is not necessarily zero. We will show that the sets of all Hurwitz automorphisms, comorphisms, and derivations of the underlying ring are naturally isomorphic to one another.

The unitary symmetric monoidal model category of small C*-categories

Wed, 12 Dec 2012 09:11 EST

Ivo Dell'Ambrogio

Source: Homology Homotopy Appl., Volume 14, Number 2, 101--127.

Abstract:
We produce a cofibrantly generated simplicial symmetric monoidal model structure for the category of (small unital) $\mathrm{C}^*$-categories, whose weak equivalences are the unitary equivalences. The closed monoidal structure consists of the maximal tensor product, which generalizes that of $\mathrm{C}^*$-algebras, together with the Ghez-Lima-Roberts $\mathrm{C}^*$-categories of *-functors, $\mathrm{C}^*(A;B)$, providing the internal Hom’s.

Weight structure on Kontsevich's noncommutative mixed motives

Wed, 12 Dec 2012 09:11 EST

Gonçalo Tabuada

Source: Homology Homotopy Appl., Volume 14, Number 2, 129--142.

Abstract:
In this article we endow Kontsevich’s triangulated category $\mathrm{KMM}_k$ of noncommutative mixed motives with a nondegenerate weight structure in the sense of Bondarko. As an application we obtain: (1) a convergent weight spectral sequence for every additive invariant (e.g., algebraic $K$-theory, cyclic homology, topological Hochschild homology, etc.); (2) a ring isomorphism between $K_0(\mathrm{KMM}_k)$ and the Grothendieck ring of the category of noncommutative Chow motives; (3) a precise relationship between Voevodsky’s (virtual) mixed motives and Kontsevich’s noncommutative (virtual) mixed motives.

The equvariant slice filtration: A primer

Wed, 12 Dec 2012 09:11 EST

Michael A. Hill

Source: Homology Homotopy Appl., Volume 14, Number 2, 143--166.

Abstract:
We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice-connectivity of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the Eilenberg- Mac Lane spectrum associated to a Mackey functor for a cyclic p-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullbacks.

Homology decompositions and groups inducing fusion systems

Wed, 12 Dec 2012 09:11 EST

Assaf Libman, Nora Seeliger

Source: Homology Homotopy Appl., Volume 14, Number 2, 167--187.

Abstract:
We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of $p$-local finite groups. We prove that the cohomology ring of Robinson’s construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.

Every binary self-dual code arises from Hilbert symbols

Wed, 12 Dec 2012 09:11 EST

Ted Chinburg, Ying Zhang

Source: Homology Homotopy Appl., Volume 14, Number 2, 189--196.

Abstract:
In this paper we construct binary self-dual codes using the étale cohomology of $\mu_2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all selfdual codes of length at least 4 arise from Hilbert pairings on rings of $S$-integers of $\mathbb{Q}$. This is an arithmetic counterpart of a result of Kreck and Puppe, who used cobordism theory to show that all self-dual codes arise from Poincaré duality on real three manifolds.

Chromatic subdivision of a simplicial complex

Wed, 12 Dec 2012 09:11 EST

Dmitry N. Kozlov

Source: Homology Homotopy Appl., Volume 14, Number 2, 197--209.

Abstract:
We prove that the protocol complex of the immediate snapshot read/write complex for $n + 1$ processors is a simplicial subdivision of the input complex. Our proof is purely geometric, using the Schlegel diagram construction.

$K$-motives of algebraic varieties

Wed, 12 Dec 2012 09:11 EST

Grigory Garkusha, Ivan Panin

Source: Homology Homotopy Appl., Volume 14, Number 2, 211--264.

Abstract:
A kind of motivic algebra of spectral categories and modules over them is developed to introduce $K$-motives of algebraic varieties. As an application, bivariant algebraic $K$-theory $K(X; Y)$ as well as bivariant motivic cohomology groups $H^{p;q}(X; Y; \mathbb{Z})$ are defined and studied. We use Grayson’s machinery to produce the Grayson motivic spectral sequence connecting bivariant $K$-theory to bivariant motivic cohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of $K$-motives constructed in the paper. It is also shown that ordinary algebraic $K$-theory is represented by the $K$-motive of the point.

Quillen model structions on the category of graphs

Wed, 12 Dec 2012 09:11 EST

Jean-Marie Droz

Source: Homology Homotopy Appl., Volume 14, Number 2, 265--284.

Abstract:
We present different ways of endowing a particular category of graphs with Quillen model structures. We show, among other things, that the core of a graph can be seen as its homotopy type in an appropriate Quillen model structure, and that an infinity of Quillen model structures exist for our particular category of graphs.