Introduction To L2 Invariants

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Introduction To L2 Invariants

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Author : Holger Kammeyer
language : en
Publisher: Springer Nature
Release Date : 2019-10-29

Introduction To L2 Invariants written by Holger Kammeyer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-29 with Mathematics categories.

This book introduces the reader to the most important concepts and problems in the field of l2-invariants. After some foundational material on group von Neumann algebras, l2-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of l2-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of l2-Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lück's approximation theorem and its generalizations. The final chapter deals with l2-torsion, twisted variants and the conjectures relating them to torsion growth in homology. The text provides a self-contained treatment that constructs the required specialized concepts from scratch. It comes with numerous exercises and examples, so that both graduate students and researchers will find it useful for self-study or as a basis for an advanced lecture course.

L2 Invariants Theory And Applications To Geometry And K Theory

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Author : Wolfgang Lück
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09

L2 Invariants Theory And Applications To Geometry And K Theory written by Wolfgang Lück and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.

In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.

Introduction To Algebraic Topology

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Author : Holger Kammeyer
language : en
Publisher: Springer Nature
Release Date : 2022-06-20

Introduction To Algebraic Topology written by Holger Kammeyer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-06-20 with Mathematics categories.

This textbook provides a succinct introduction to algebraic topology. It follows a modern categorical approach from the beginning and gives ample motivation throughout so that students will find this an ideal first encounter to the field. Topics are treated in a self-contained manner, making this a convenient resource for instructors searching for a comprehensive overview of the area. It begins with an outline of category theory, establishing the concepts of functors, natural transformations, adjunction, limits, and colimits. As a first application, van Kampen's theorem is proven in the groupoid version. Following this, an excursion to cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology is then defined, motivating the Eilenberg-Steenrod axioms, and the simplicial approximation theorem is proven. After verifying the axioms for singular homology, various versions of the Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of spheres are classified by degree.The final chapter discusses cellular homology of CW complexes, culminating in the uniqueness theorem for ordinary homology. Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included.

L2 Invariants Theory And Applications To Geometry And K Theory

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Author : Wolfgang Lück
language : en
Publisher: Springer
Release Date : 2002-08-06

L2 Invariants Theory And Applications To Geometry And K Theory written by Wolfgang Lück and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-08-06 with Mathematics categories.

In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.

L2 Invariants

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Author : Wolfgang Luck
language : en
Publisher:
Release Date : 2014-01-15

L2 Invariants written by Wolfgang Luck and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.

Ergodic Theoretic Methods In Group Homology

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Author : Clara Löh
language : en
Publisher: Springer Nature
Release Date : 2020-03-14

Ergodic Theoretic Methods In Group Homology written by Clara Löh and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-14 with Mathematics categories.

This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers. Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds. This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school “Random and arithmetic structures in topology” and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.

L2 Invariants For Quantum Groups

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Author :
language : en
Publisher:
Release Date : 2008

L2 Invariants For Quantum Groups written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with categories.

Introduction To Vassiliev Knot Invariants

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Author : S. Chmutov
language : en
Publisher: Cambridge University Press
Release Date : 2012-05-24

Introduction To Vassiliev Knot Invariants written by S. Chmutov and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-24 with Mathematics categories.

A detailed exposition of the theory with an emphasis on its combinatorial aspects.

Mathematical Survey Lectures 1943 2004

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Author : Beno Eckmann
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-10-11

Mathematical Survey Lectures 1943 2004 written by Beno Eckmann and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-10-11 with Mathematics categories.

This collection of survey lectures in mathematics traces the career of Beno Eckmann, whose work ranges across a broad spectrum of mathematical concepts from topology through homological algebra to group theory. One of our most influential living mathematicians, Eckmann has been associated for nearly his entire professional life with the Swiss Federal Technical University (ETH) at Zurich, as student, lecturer, professor, and professor emeritus.

Trends In Contemporary Mathematics

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Author : Vincenzo Ancona
language : en
Publisher: Springer
Release Date : 2014-08-27

Trends In Contemporary Mathematics written by Vincenzo Ancona and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-08-27 with Mathematics categories.

The topics faced in this book cover a large spectrum of current trends in mathematics, such as Shimura varieties and the Lang lands program, zonotopal combinatorics, non linear potential theory, variational methods in imaging, Riemann holonomy and algebraic geometry, mathematical problems arising in kinetic theory, Boltzmann systems, Pell's equations in polynomials, deformation theory in non commutative algebras. This work contains a selection of contributions written by international leading mathematicians who were speakers at the "INdAM Day", an initiative born in 2004 to present the most recent developments in contemporary mathematics.