A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

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.     

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.     

Analytical solution of variant Boussinesq equations

In this paper, analytic solutions of the variant Boussinesq equations are obtained by the homotopy analysis and the homotopy Pad methods. The obtained approximation using homotopy method contains an auxiliary parameter, which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Pad technique is often independent of auxiliary parameter , and this technique accelerates the convergence of the related series. Copyright © 2013 John Wiley & Sons, Ltd.     

Infima of universal energy functionals on homotopy classes

We consider the infima (f) on homotopy classes of energy functionals E defined on smooth maps f: Mⁿ → V^k between compact connected Riemannian manifolds. If M contains a sub‐manifold L of codimension greater than the degree of E then (f) is determined by the homotopy class of the restriction of f to M \ L. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 .     

球面稳定同伦群中的一族新元素

本文研究了球面稳定同伦群中元素的非平凡性.利用May谱序列,证明了在Adams谱序列E₂项中存在乘积元素收敛到球面稳定同伦群的一族阶为p的非零元,此非零元具有更高维数的滤子.     

Detection of Some Elements in the Stable Homotopy Groups of Spheres

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π＊S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres πp^nq＋2pq＋q-3S which isof order p and is represented by kohn ∈ ExtA^3,P^nq＋2pq＋q（Zp,Zp） in the Adams spectral sequence, wherep 〉 5 is an odd prime, n ≥3 and q = 2（p-1）. In the course of the proof, a new family of homotopy elements in πp^nq＋（p＋1）q-1V（1） which is represented by β＊i＇＊i＊（hn） ∈ ExtA^2,pnq＋（p＋1）q＋1 （H^＊V（1）, Zp） in the Adams sequence is detected.     

Free Group Actions on Spaces Homotopy Equivalent to a Sphere

For infinite-dimensional homotopy space forms X/G with G nontrivial finite cyclic groups, we study the homotopy type of X/G. We show that the Euler class of X/G is not zero if and only if X/G is finitely dominated.     

The classifying space of a categorical crossed module

Any pointed CW‐complex X has associated a categorical crossed module WX whose homotopy groups coincide with those of the space up to dimension 3. Here we associate WX more closely with the homotopy 3‐type of X. We introduce the nerve of a categorical crossed module L and define its classifying space BL as the geometrical realization of the nerve. Then we prove that there is a map X → BWX inducing isomorphism of the homotopy groups π_i for i ≤ 3. Finally, comparison with other algebraic models of 3‐types is achieved (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.     

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.     

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.     

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.     

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.     

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     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Homotopy invariance of Novikov-Shubin invariants and Betti numbers

We give short proofs of the Gromov-Shubin theorem on the homotopy invariance of the Novikov-Shubin invariants and of the Dodziuk theorem on the homotopy invariance of the Betti numbers of the universal covering of a closed manifold in this paper. We show that the homotopy invariance of these invariants is no more difficult to prove than the homotopy invariance of ordinary homology theory.

     

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.     

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.