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Softcover ISBN:  9781470466749 
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Softcover ISBN:  9781470466749 
Product Code:  GSM/218.S 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470466732 
EPUB ISBN:  9781470469511 
Product Code:  GSM/218.E 
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MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470466749 
eBook ISBN:  9781470466732 
Product Code:  GSM/218.S.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 

Book DetailsGraduate Studies in MathematicsVolume: 218; 2021; 425 ppMSC: Primary 58; 55; 57
This book gives a comprehensive introduction to the theory of smooth manifolds, maps, and fundamental associated structures with an emphasis on “bare hands” approaches, combining differentialtopological cutandpaste procedures and applications of transversality. In particular, the smooth cobordism cupproduct is defined from scratch and used as the main tool in a variety of settings. After establishing the fundamentals, the book proceeds to a broad range of more advanced topics in differential topology, including degree theory, the PoincaréHopf index theorem, bordismcharacteristic numbers, and the PontryaginThom construction. Cobordism intersection forms are used to classify compact surfaces; their quadratic enhancements are developed and applied to studying the homotopy groups of spheres, the bordism group of immersed surfaces in a 3manifold, and congruences mod 16 for the signature of intersection forms of 4manifolds. Other topics include the highdimensional \(h\)cobordism theorem stressing the role of the “Whitney trick”, a determination of the singleton bordism modules in low dimensions, and proofs of parallelizability of orientable 3manifolds and the LickorishWallace theorem. Nash manifolds and Nash's questions on the existence of real algebraic models are also discussed.
This book will be useful as a textbook for beginning masters and doctoral students interested in differential topology, who have finished a standard undergraduate mathematics curriculum. It emphasizes an active learning approach, and exercises are included within the text as part of the flow of ideas. Experienced readers may use this book as a source of alternative, constructive approaches to results commonly presented in more advanced contexts with specialized techniques.
ReadershipGraduate students interested in a graduate level introduction to differential topology.

Table of Contents

Chapters

The smooth category of open subsets of Euclidean spaces

The category of embedded smooth manifolds

Stiefel and Grassmann manifolds

The category of smooth manifolds

Tautological bundles and pullback

Compact embedded smooth manifolds

Cut and paste compact manifolds

Transversality

Morse functions and handle decompositions

Bordism

Smooth cobordism

Applications of cobordism rings

Line bundles, hypersurfaces, and cobordism

EulerPoincaré characteristic

Surfaces

Bordism characteristic numbers

The PontryaginThom construction

Highdimensional manifolds

On 3manifolds

On 4manifolds

Baby categories


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This book gives a comprehensive introduction to the theory of smooth manifolds, maps, and fundamental associated structures with an emphasis on “bare hands” approaches, combining differentialtopological cutandpaste procedures and applications of transversality. In particular, the smooth cobordism cupproduct is defined from scratch and used as the main tool in a variety of settings. After establishing the fundamentals, the book proceeds to a broad range of more advanced topics in differential topology, including degree theory, the PoincaréHopf index theorem, bordismcharacteristic numbers, and the PontryaginThom construction. Cobordism intersection forms are used to classify compact surfaces; their quadratic enhancements are developed and applied to studying the homotopy groups of spheres, the bordism group of immersed surfaces in a 3manifold, and congruences mod 16 for the signature of intersection forms of 4manifolds. Other topics include the highdimensional \(h\)cobordism theorem stressing the role of the “Whitney trick”, a determination of the singleton bordism modules in low dimensions, and proofs of parallelizability of orientable 3manifolds and the LickorishWallace theorem. Nash manifolds and Nash's questions on the existence of real algebraic models are also discussed.
This book will be useful as a textbook for beginning masters and doctoral students interested in differential topology, who have finished a standard undergraduate mathematics curriculum. It emphasizes an active learning approach, and exercises are included within the text as part of the flow of ideas. Experienced readers may use this book as a source of alternative, constructive approaches to results commonly presented in more advanced contexts with specialized techniques.
Graduate students interested in a graduate level introduction to differential topology.

Chapters

The smooth category of open subsets of Euclidean spaces

The category of embedded smooth manifolds

Stiefel and Grassmann manifolds

The category of smooth manifolds

Tautological bundles and pullback

Compact embedded smooth manifolds

Cut and paste compact manifolds

Transversality

Morse functions and handle decompositions

Bordism

Smooth cobordism

Applications of cobordism rings

Line bundles, hypersurfaces, and cobordism

EulerPoincaré characteristic

Surfaces

Bordism characteristic numbers

The PontryaginThom construction

Highdimensional manifolds

On 3manifolds

On 4manifolds

Baby categories