Author : L. Gaunce Lewis
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 20,39 MB
Release : 2000
Category : Mathematics
ISBN : 082182046X
This book is intended for graduate students and research mathematicians interested in algebraic topology.
Author :
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 41,99 MB
Release : 2000-03-03
Category : Mathematics
ISBN : 9780821864098
Let $G$ be a compact Lie group, $\Pi$ be a normal subgroup of $G$, $\mathcal G=G/\Pi$, $X$ be a $\mathcal G$-space and $Y$ be a $G$-space. There are a number of results in the literature giving a direct sum decomposition of the group $[\Sigma^\infty X,\Sigma^\infty Y]_G$ of equivariant stable homotopy classes of maps from $X$ to $Y$. Here, these results are extended to a decomposition of the group $[B,C]_G$ of equivariant stable homotopy classes of maps from an arbitrary finite $\mathcal G$-CW sptrum $B$ to any $G$-spectrum $C$ carrying a geometric splitting (a new type of structure introduced here). Any naive $G$-spectrum, and any spectrum derived from such by a change of universe functor, carries a geometric splitting. Our decomposition of $[B,C]_G$ is a consequence of the fact that, if $C$ is geometrically split and $(\mathfrak F',\mathfrak F)$ is any reasonable pair of families of subgroups of $G$, then there is a splitting of the cofibre sequence $(E\mathfrak F_+ \wedge C)^\Pi \longrightarrow (E\mathfrak F'_+ \wedge C)^\Pi \longrightarrow (E(\mathfrak F', \mathfrak F) \wedge C)^\Pi$ constructed from the universal spaces for the families. Both the decomposition of the group $[B,C]_G$ and the splitting of the cofibre sequence are proven here not just for complete $G$-universes, but for arbitrary $G$-universes. Various technical results about incomplete $G$-universes that should be of independent interest are also included in this paper. These include versions of the Adams and Wirthmuller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed-point spectrum $(E(\mathfrak F',\mathfrak F) \wedge C)^\Pi$ which gives computational force to the intuition that what really matters about a $G$-universe $U$ is which orbits $G/H$ embed as $G$-spaces in $U$.
Author : Erik Guentner
Publisher : American Mathematical Soc.
Page : 101 pages
File Size : 16,75 MB
Release : 2000
Category : Mathematics
ISBN : 0821821164
This title examines the equivariant e-theory for c*-algebra, focusing on research carried out by Higson and Kasparov. Let A and B be C*-algebras which are equipped with continuous actions of a second countable, locally compact group G. We define a notion of equivariant asymptotic morphism, and use it to define equivariant E-theory groups EULG(A, B) which generalize the E-theory groups of Connes and Higson. We develop the basic properties of equivariant E-theory, including a composition product and six-term exact sequences in both variables, and apply our theory to the problem of calculating K-theory for group C*-algebras. Our main theorem gives a simple criterion for the assembly map of Baum and Connes to be an isomorphism. The result plays an important role in the work of Higson and Kasparov on the Bau m-Connes conjecture for groups which act isometrically and metrically properly on Hilbert space
Author : Laura Ann Smithies
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 17,96 MB
Release : 2001
Category : Mathematics
ISBN : 0821827251
This book is intended for graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.
Author : Henning Krause
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 16,40 MB
Release : 2001
Category : Mathematics
ISBN : 0821826182
These notes present an introduction into the spectrum of the category of modules over a ring. We discuss the general theory of pure-injective modules and concentrate on the isomorphism classes of indecomposable pure-injective modules which form the underlying set of this spectrum. The interplay between the spectrum and the category of finitely presented modules provides new insight into the geometrical and homological properties of the category of finitely presented modules. Various applications from representation theory of finite dimensional algebras are included.
Author : Vitaly Bergelson
Publisher : American Mathematical Soc.
Page : 121 pages
File Size : 34,31 MB
Release : 2000
Category : Mathematics
ISBN : 0821826573
The authors prove a polynomial multiple recurrence theorem for finitely many commuting measure preserving transformations of a probability space, extending a polynomial Szemerédi theorem appearing in [BL1]. The linear case is a consequence of an ergodic IP-Szemerédi theorem of Furstenberg and Katznelson ([FK2]). Several applications to the fine structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which we also prove a multiparameter weakly mixing polynomial ergodic theorem. The techniques and apparatus employed include a polynomialization of an IP structure theory developed in [FK2], an extension of Hindman's theorem due to Milliken and Taylor ([M], [T]), a polynomial version of the Hales-Jewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IP-systems of unitary operators ([BFM]).
Author : M. A. Dickmann
Publisher : American Mathematical Soc.
Page : 271 pages
File Size : 41,2 MB
Release : 2000
Category : Mathematics
ISBN : 0821820575
This monograph presents a systematic study of Special Groups, a first-order universal-existential axiomatization of the theory of quadratic forms, which comprises the usual theory over fields of characteristic different from 2, and is dual to the theory of abstract order spaces. The heart of our theory begins in Chapter 4 with the result that Boolean algebras have a natural structure of reduced special group. More deeply, every such group is canonically and functorially embedded in a certain Boolean algebra, its Boolean hull. This hull contains a wealth of information about the structure of the given special group, and much of the later work consists in unveiling it. Thus, in Chapter 7 we introduce two series of invariants "living" in the Boolean hull, which characterize the isometry of forms in any reduced special group. While the multiplicative series--expressed in terms of meet and symmetric difference--constitutes a Boolean version of the Stiefel-Whitney invariants, the additive series--expressed in terms of meet and join--, which we call Horn-Tarski invariants, does not have a known analog in the field case; however, the latter have a considerably more regular behaviour. We give explicit formulas connecting both series, and compute explicitly the invariants for Pfister forms and their linear combinations. In Chapter 9 we combine Boolean-theoretic methods with techniques from Galois cohomology and a result of Voevodsky to obtain an affirmative solution to a long standing conjecture of Marshall concerning quadratic forms over formally real Pythagorean fields. Boolean methods are put to work in Chapter 10 to obtain information about categories of special groups, reduced or not. And again in Chapter 11 to initiate the model-theoretic study of the first-order theory of reduced special groups, where, amongst other things we determine its model-companion. The first-order approach is also present in the study of some outstanding classes of morphisms carried out in Chapter 5, e.g., the pure embeddings of special groups. Chapter 6 is devoted to the study of special groups of continuous functions.
Author : Robert Rumely
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 47,81 MB
Release : 2000
Category : Mathematics
ISBN : 0821820583
In the case where the norms are induced by metrics on the fibres of ${\mathcal L}$, we establish the functoriality of the sectional capacity under base change, pullbacks by finite surjective morphisms, and products. We study the continuity of $S Gamma(\overline{\mathcal L})$ under variation of the metric and line bundle, and we apply this to show that the notion of $v$-adic sets in $X(\mathbb C v)$ of capacity $0$ is well-defined. Finally, we show that sectional capacities for arbitrary norms can be well-approximated using objects of finite type.
Author : Bernhard Lani-Wayda
Publisher : American Mathematical Soc.
Page : 138 pages
File Size : 28,12 MB
Release : 2001
Category : Mathematics
ISBN : 0821826808
This book is intended for graduate students and research mathematicians interested in mechanics of particle systems.
Author : Piotr Hajłasz
Publisher : American Mathematical Soc.
Page : 119 pages
File Size : 11,10 MB
Release : 2000
Category : Mathematics
ISBN : 0821820478
There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Caratheodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space $X$ equipped with a doubling measure $\mu$. A generalization of a Sobolev function and its gradient is a pair $u\in L^{1}_{\rm loc}(X)$, $0\leq g\in L^{p}(X)$ such that for every ball $B\subset X$ the Poincare-type inequality $ \intbar_{B} u-u_{B} \, d\mu \leq C r ( \intbar_{\sigma B} g^{p}\, d\mu)^{1/p}\,$ holds, where $r$ is the radius of $B$ and $\sigma\geq 1$, $C>0$ are fixed constants. Working in the above setting we show that basically all relevant results from the classical theory have their counterparts in our general setting. These include Sobolev-Poincare type embeddings, Rellich-Kondrachov compact embedding theorem, and even a version of the Sobolev embedding theorem on spheres. The second part of the paper is devoted to examples and applications in the above mentioned areas.
