Boundedly Controlled Topology

Boundedly Controlled Topology PDF

Author: Douglas R. Anderson

Publisher: Springer

Published: 2006-11-15

Total Pages: 322

ISBN-13: 3540392491

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Several recent investigations have focused attention on spaces and manifolds which are non-compact but where the problems studied have some kind of "control near infinity". This monograph introduces the category of spaces that are "boundedly controlled" over the (usually non-compact) metric space Z. It sets out to develop the algebraic and geometric tools needed to formulate and to prove boundedly controlled analogues of many of the standard results of algebraic topology and simple homotopy theory. One of the themes of the book is to show that in many cases the proof of a standard result can be easily adapted to prove the boundedly controlled analogue and to provide the details, often omitted in other treatments, of this adaptation. For this reason, the book does not require of the reader an extensive background. In the last chapter it is shown that special cases of the boundedly controlled Whitehead group are strongly related to lower K-theoretic groups, and the boundedly controlled theory is compared to Siebenmann's proper simple homotopy theory when Z = IR or IR2.

Boundedly Controlled Topology

Boundedly Controlled Topology PDF

Author: Douglas Ross Anderson

Publisher: Springer

Published: 1988

Total Pages: 340

ISBN-13:

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Several recent investigations have focused attention on spaces and manifolds which are non-compact but where the problems studied have some kind of "control near infinity". This monograph introduces the category of spaces that are "boundedly controlled" over the (usually non-compact) metric space Z. It sets out to develop the algebraic and geometric tools needed to formulate and to prove boundedly controlled analogues of many of the standard results of algebraic topology and simple homotopy theory. One of the themes of the book is to show that in many cases the proof of a standard result can be easily adapted to prove the boundedly controlled analogue and to provide the details, often omitted in other treatments, of this adaptation. For this reason, the book does not require of the reader an extensive background. In the last chapter it is shown that special cases of the boundedly controlled Whitehead group are strongly related to lower K-theoretic groups, and the boundedly controlled theory is compared to Siebenmann's proper simple homotopy theory when Z = IR or IR2.

Control and Relaxation over the Circle

Control and Relaxation over the Circle PDF

Author: Bruce Hughes

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 113

ISBN-13: 0821820699

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This work formulates and proves a geometric version of the fundamental theorem of algebraic K-theory which relates the K-theory of the Laurent polynomial extension of a ring to the K-theory of the ring. The geometric version relates the higher simple homotopy theory of the product of a finite complex and a circle with that of the complex. By using methods of controlled topology, we also obtain a geometric version of the fundamental theorem of lower algebraic K-theory. The main new innovation is a geometrically defined nil space.

The Topological Classification of Stratified Spaces

The Topological Classification of Stratified Spaces PDF

Author: Shmuel Weinberger

Publisher: University of Chicago Press

Published: 1994

Total Pages: 314

ISBN-13: 9780226885667

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This book provides the theory for stratified spaces, along with important examples and applications, that is analogous to the surgery theory for manifolds. In the first expository account of this field, Weinberger provides topologists with a new way of looking at the classification theory of singular spaces with his original results. Divided into three parts, the book begins with an overview of modern high-dimensional manifold theory. Rather than including complete proofs of all theorems, Weinberger demonstrates key constructions, gives convenient formulations, and shows the usefulness of the technology. Part II offers the parallel theory for stratified spaces. Here, the topological category is most completely developed using the methods of "controlled topology." Many examples illustrating the topological invariance and noninvariance of obstructions and characteristic classes are provided. Applications for embeddings and immersions of manifolds, for the geometry of group actions, for algebraic varieties, and for rigidity theorems are found in Part III. This volume will be of interest to topologists, as well as mathematicians in other fields such as differential geometry, operator theory, and algebraic geometry.

Geometry and Topology

Geometry and Topology PDF

Author: Mccrory

Publisher: CRC Press

Published: 2020-12-17

Total Pages: 366

ISBN-13: 1000110842

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This book discusses topics ranging from traditional areas of topology, such as knot theory and the topology of manifolds, to areas such as differential and algebraic geometry. It also discusses other topics such as three-manifolds, group actions, and algebraic varieties.

Geometry and Topology

Geometry and Topology PDF

Author: Mccrory

Publisher: CRC Press

Published: 1986-10-22

Total Pages: 370

ISBN-13: 9780824776213

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This book discusses topics ranging from traditional areas of topology, such as knot theory and the topology of manifolds, to areas such as differential and algebraic geometry. It also discusses other topics such as three-manifolds, group actions, and algebraic varieties.

Recursion Theory Week

Recursion Theory Week PDF

Author: Klaus Ambos-Spies

Publisher: Springer

Published: 2006-11-14

Total Pages: 398

ISBN-13: 3540471421

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These proceedings contain research and survey papers from many subfields of recursion theory, with emphasis on degree theory, in particular the development of frameworks for current techniques in this field. Other topics covered include computational complexity theory, generalized recursion theory, proof theoretic questions in recursion theory, and recursive mathematics.

Groups of Self-Equivalences and Related Topics

Groups of Self-Equivalences and Related Topics PDF

Author: Renzo A. Piccinini

Publisher: Springer

Published: 2006-11-14

Total Pages: 223

ISBN-13: 3540470913

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Since the subject of Groups of Self-Equivalences was first discussed in 1958 in a paper of Barcuss and Barratt, a good deal of progress has been achieved. This is reviewed in this volume, first by a long survey article and a presentation of 17 open problems together with a bibliography of the subject, and by a further 14 original research articles.

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