Long Box Bracket Operations in Homotopy Theory

Secondary homotopy operations called box bracket operations were defined in the homotopy theory of an arbitrary 2-category with zeros by Hardie, Marcum and Oda (Rend Ist Mat Univ Trieste, 33:19–70 2001). For the topological 2-category of based spaces, based maps and based track classes of based homotopies, the classical Toda bracket is a particular example of a box bracket operation and subsequent development of the theory has refined, clarified and placed in this more general context many of the properties of classical Toda brackets. In this paper, and for the topological case only, we use an inductive definition to extend the theory to long box brackets. As is well-known, the necessity to manage higher homotopy coherence is a complicating factor in the consideration of such higher order operations. The key to our construction is the definition of an appropriate triple box bracket operation and consequently we focus primarily on the properties of the triple box bracket. We exhibit and exploit the relationship of the classical quaternary Toda bracket to the triple box bracket. As our main results we establish some computational techniques for triple box brackets that are based on composition methods. Some specific computations from the homotopy groups of spheres are included.     

TODA BRACKETS AND THE CATEGORY OF HOMOTOPY PAIRS

A new treatment is given of the cylinder-web diagram and associated diagonal sequences in homotopy pair theory. The efficiency of the diagram as a machine for computing homotopy pair groups is enhanced by a result that traces the path of a Toda bracket element through the arrows of the diagram. The diagonal factorization problem for a homotopy pair class is studied and related to the behaviour of Toda brackets. A necessary and sufficient condition for the vanishing of a Toda bracket is obtained.     

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.     

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X ₀ be a topological component of a nonsingular real algebraic variety and i:X → X _C is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X ₀ → X _C and obtain several results using rational homotopy theory and other standard tools of homotopy theory.     

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.     

Commuting Homotopy Limits and Smash Products

In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.     

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

Higher homotopy operations

We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. Second author partially supported by grant GA AV R 1019203.     

A 2-groupoid Characterisation of the Cubical Homotopy Pushout

We give a 2-track-theoretical characterisation of the homotopy pushout of a 3-corner by recognising the mapping 2-simplex as an initial object in a coherent homotopy category of Hausdorff spaces under a 3-corner with morphisms expressed in terms of the 1-morphisms and 2-morphisms of a homotopy 2-groupoid.     

The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.      

Homotopy Structures for Algebras over a Monad

Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A A, or path-endofunctor, can be lifted to the category A ^T of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation : T P P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of algebras for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the homotopy operations (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to A ^T , just by verifying the consistency between these operations and : T P P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy.     

Hom-Lie quadratic and Pinczon Algebras

Presenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.     

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 .     

The coherent homotopy category over a fixed space is a category of fractions

In this note we prove that the coherent homotopy category over a fixed space B with morphisms represented by certain homotopy commutative squares (see [8]) is isomorphic to the category obtained by formally inverting those maps in the category Top_B of topological spaces over B which are ordinary homotopy equivalences.     

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a system of open neighbourhoods at infinity while an exterior map is a continuous map which is continuous at infinity. The category of spaces and proper maps is a subcategory of the category of exterior spaces.In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T,T} of suitable exterior spaces. For this goal, for a given exterior space T, we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown–Grossman, erin–Steenrod, p-cylindrical, Baues–Quintero and Farrell–Taylor–Wagoner groups, as well as the standard (Hurewicz) homotopy groups.The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T,T}, it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Derived brackets and sh Leibniz algebras

We develop a general framework for the construction of various derived brackets. We show that suitably deforming the differential of a graded Leibniz algebra extends the derived bracket construction and leads to the notion of strong homotopy (sh) Leibniz algebra. We discuss the connections among homotopy algebra theory, deformation theory and derived brackets. We prove that the derived bracket construction induces a map from suitably defined deformation theory equivalence classes to the isomorphism classes of sh Leibniz algebras.     

2-Groupoid Enrichments in Homotopy Theory and Algebra

The use of groupoid enrichments in abstract homotopy theory is well known and classical. Recently enrichments by higher-dimensional groupoids have been considered. Here we will describe enrichment by 2-groupoids with respect to the Gray tensor product and will examine several examples (2-groupoids, 2-crossed complexes, chain complexes, etc.) from an elementary view-point. The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.     

A Coherent Homotopy Category of 2-track Commutative Cubes

We consider a category \({\mathcal H}^{\ominus \otimes}\) (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of \({\mathcal H}^{\ominus \otimes}\) to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation.     

Homotopy operations and the obstructions to being anH-space

We describe an obstruction theory for a given topological spaceX to be anH-space, in terms of higher homotopy operations and show how this theory can be used to calculate such operations in certain cases. Date: September 13, 1995