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Algebra and Applications #17: Syzygies and Homotopy Theory

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Algebra and Applications #17: Syzygies and Homotopy Theory Cover

 

Synopses & Reviews

Publisher Comments:

The most important invariant of a topological space is its fundamental group. When this is trivial, the resulting homotopy theory is well researched and familiar. In the general case, however, homotopy theory over nontrivial fundamental groups is much more problematic and far less well understood.Syzygies and Homotopy Theory explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents, for the first time, a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory. The first part of the book is theoretical, formulated to allow a general finitely presented group as a fundamental group. The innovation here is to regard syzygies as stable modules rather than minimal modules. Inevitably this forces a reconsideration of the problems of noncancellation; these are confronted in the second, practical, part of the book. In particular, the second part of the book considers how the theory works out in detail for the specific examples Fn ´F where Fn is a free group of rank n and F is finite. Another innovation is to parametrize the first syzygy in terms of the more familiar class of stably free modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares. The theory developed within this book has potential applications in various branches of algebra, including homological algebra, ring theory and K-theory. Syzygies and Homotopy Theory will be of interest to researchers and also to graduate students with a background in algebra and algebraic topology.

Synopsis:

This book explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory.

Table of Contents

Preliminaries.- The restricted linear group.- The calculus of corners and squares.- Extensions of modules.- The derived module category.- Finiteness conditions.- The Swan mapping.- Classification of algebraic complexes.- Rings with stably free cancellation.- Group rings of cyclic groups.- Group rings of dihedral groups.- Group rings of quaternionic groups.- Parametrizing W1 (Z) : generic case.- Parametrizing W1 (Z) : singular case.- Generalized Swan modules.- Parametrizing W1 (Z) : G = C¥ ´ F.- Conclusion​.

Product Details

ISBN:
9781447122937
Author:
Johnson, F. E. A.
Publisher:
Springer
Author:
Johnson, F. E. a.
Subject:
Algebra - Abstract
Subject:
D(2) problem
Subject:
Milnor squares
Subject:
R(2) problem
Subject:
generalized Swan module
Subject:
stable module
Subject:
syzygy
Subject:
Group Theory and Generalizations
Subject:
Commutative Rings and Algebras
Subject:
Associative Rings and Algebras
Subject:
Category Theory, Homological Algebra
Subject:
Manifolds and Cell Complexes (incl. Diff.Topology)
Subject:
Mathematics-Abstract Algebra
Copyright:
Edition Description:
2012
Series:
Algebra and Applications
Series Volume:
17
Publication Date:
20111215
Binding:
HARDCOVER
Language:
English
Pages:
318
Dimensions:
235 x 155 mm

Related Subjects

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Science and Mathematics » Mathematics » Algebra » Abstract Algebra
Science and Mathematics » Mathematics » Topology

Algebra and Applications #17: Syzygies and Homotopy Theory New Hardcover
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Product details 318 pages Springer - English 9781447122937 Reviews:
"Synopsis" by , This book explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory.
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