The odd primary -structure of low rank Lie groups and its application to exponents

A compact, connected, simple Lie group localized at an odd prime is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of is low. This holds for , for example, if . The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to . This is applied to prove useful information about the torsion in the homotopy groups of , including an upper bound on its exponent.

     

Anick's spaces and the double loops of odd primary Moore spaces

Several properties of Anick's spaces are established which give a retraction of Anick's off if and . The proof is alternate to and more immediate than the two proofs of Neisendorfer's.

     

Divisor spaces on punctured Riemann surfaces

In this paper, we study the topology of spaces of -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163-221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.

     

Interpolation theorems for variable exponent Lebesgue spaces

When Hardy-Littlewood maximal operator is bounded on Lp(⋅)(Rn) space we prove θ[Lp(⋅)(Rn),BMO(Rn)]=Lq(⋅)(Rn) where q(⋅)=p(⋅)/(1−θ) and θ[Lp(⋅)(Rn),H¹(Rn)]=Lq(⋅)(Rn) where 1/q(⋅)=θ+(1−θ)/p(⋅).     

Actor of a Precrossed Module

Applying the equivalence of the category of precrossed modules with the category of groups with two additional unary operations satisfying the corresponding conditions, the construction of an actor is given in terms of Whitehead group of generalized regular derivations, defined in the article, and the automorphism group of a precrossed module. The analogous approach to this problem in the case of crossed modules leads to the well-known construction given in the works of Lue and Norrie.     

多线性Hardy型算子在变指数Herz-Morrey乘积空间上的有界性

本文证明了多线性分数次Hardy算子Hβ,m和H *β,m (m∈Z+且m≥1)在变指数Herz-Morrey乘积空间上的有界性.对多线性Hardy算子也建立了相应的结果.     

Topological structure of Urysohn universal spaces

The main aim of the paper is to prove that every nonempty member P of the algebra of subsets of a nontrivial Urysohn space generated by all balls (open and closed) is an l₂-manifold of finite homotopy type. An algorithm of finding a polyhedron K such that P and K×l₂ are homeomorphic is presented. An alternative proof of the Uspenskij theorem [V.V. Uspenskij, The Urysohn universal metric space is homeomorphic to a Hilbert space, Topology Appl. 139 (2004) 145-149] is given.     

Function spaces and product topologies on powers of spaces

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.     

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex the following conditions are equivalent:

• iff for any space .
• There exists a -invertible map of a metrizable compactum with onto the Hilbert cube.

     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.      

Universal uniform Eberlein compact spaces

A universal space is one that continuously maps onto all others of its own kind and weight. We investigate when a universal Uniform Eberlein compact space exists for weight . If , then they exist whereas otherwise, in many cases including , it is consistent that they do not exist. This answers (for many and consistently for all ) a question of Benyamini, Rudin and Wage of 1977.

     

Moore空间P~n(8)及其同伦群

令G是一Abel群,m≥2是一整数．一个型为(G,m)的Moore空间是一单连通的CW-复形X,使得■_i(X)=G(i=m),0(i≠m)．这里J_i为X的第i个整系数的约化同调群．众所周知,Moore空间存在,且任何两个型为(G,n)的Moore空间有相同伦型．取G=Z_k(模k的剩余类加群)．p~n(k)=S~(n-1)∪_(kl_(n-1))e~n为型为(Z_k,n-1)的Moore空间．特别地,考虑k=8,决定了Moore空间p~n(8)的一些同伦群．主要证明工具是Toda引进的复合工具-Toda积,Gray的关于从p~n(8)到n维球面S~n的pinchin映射的同伦纤维的胞腔结构,以及关于亚稳定相对同伦群π_k(X,A)的同伦切割定理,其中A为维数小于n-1的有限CW-复形,X=A∪e~n．     

A remark on A-weakly infinite-dimensional spaces

It is shown that the product of two A-weakly infinite-dimensional spaces may fail to have this property and that under CH there is no universal space in the class of all metrizable separable A-weakly infinite-dimensional spaces.     

Gagliardo-Nirenberg type inequality for variable exponent Lebesgue spaces

We prove analogies of the classical Gagliardo-Nirenberg inequalities

     

Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c₀ of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X−X has finite Assouad dimension.     

Properties of β-connected spaces

Many characterizations of semipreconnected (=β-connected) spaces are obtained by Aho and Nieminen [2]. In this paper, we obtain further characterizations of semipreconnected spaces. We introduce β-separated and β_s-connected sets and by using these sets we obtain several properties of β-connected subspaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Properties of Θs-US spaces

Wilansky [6] introduced the notion of US spaces. It is the purpose of this paper to introduce and investigate the notion of Θs-US spaces.