###### 4-Manifolds and Kirby Calculus (Graduate Studies in

Four Manifolds

The past two decades have brought explosive growth in 4-manifold theory. Many books are currently appearing that approach the topic from viewpoints such as gauge theory or algebraic geometry. This volume, however, offers an exposition from a topological point of view. It bridges the gap to other disciplines and presents classical but important topological techniques that have not previously appeared in the literature. Part I of the text presents the basics of the theory at the second-year graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4-manifolds. It is both elementary and comprehensive. Part III offers in depth a broad range of topics from current 4-manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. Applications are featured, and there are over 300 illustrations and numerous exercises with solutions in the book.

- Brand: Brand: American Mathematical Society
- ASIN: 0821809946

###### The Geometry of Four-Manifolds (Oxford Mathematical

Four Manifolds

The last ten years have seen rapid advances in the understanding of differentiable four-manifolds, not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics, and have proven to be a mainspring of current mathematical research. This book provides a lucid and accessible account of the modern study of the geometry of four-manifolds. Consequently, it will be required reading for all those mathematicians and theoretical physicists whose research touches on this topic. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology--notably the definition of new invariants of four-manifolds--and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proven in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topo...

- Brand: Brand: Oxford University Press, USA
- ASIN: 0198502699 Part No: line figures

###### The Wild World of 4-Manifolds

Four Manifolds

"The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it." -- Robion C. Kirby, University of California Berkeley This is a panorama of the topology of simply-connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but too small to have room to undo them. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. The first part of the book puts things in context with a survey of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold--the intersection form--and its interaction with the topology of the manifold. The third part reviews complex surfaces as an important source of examples. The fourth and final part of the book presents gauge theory. This differential-geometric method has brought to light the unwieldy nature of smooth 4-manifolds; and although the method brings new insights, it has raised more questions than answers. The structure of the book is modular and organized into a main track of approximately 20...

- Brand: Brand: American Mathematical Society
- ASIN: 0821837494 Part No: Illustrations

###### Instantons and Four-Manifolds (Mathematical Sciences Research Institute

Four Manifolds

From the reviews of the first edition: "This book exposes the beautiful confluence of deep techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional manifolds, compact spaces locally modeled on the world in which we live and operate... The book is filled with insightful remarks, proofs, and contributions that have never before appeared in print. For anyone attempting to understand the work of Donaldson and the applications of gauge theories to four-dimensional topology, the book is a must." #Science#1 "I would strongly advise the graduate student or working mathematician who wishes to learn the analytic aspects of this subject to begin with Freed and Uhlenbeck's book." #Bulletin of the American Mathematical Society#2

- ASIN: 1461397057 Part No: 19304777

###### Gauge Theory and the Topology of Four-Manifolds (Ias/Park City Mathematics

Four Manifolds

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space. One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

- Brand: Brand: Amer Mathematical Society
- ASIN: 0821805916

###### Topology of 4-Manifolds.

Four Manifolds

One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topological development of this subject, proving the Poincar and Annulus conjectures respectively. Recognition for this work includes the award of the Fields Medal of the International Congress of Mathematicians to Freedman in 1986. In Topology of 4-Manifolds these authors have collaborated to give a complete and accessible account of the current state of knowledge in this field. The basic material has been considerably simplified from the original publications, and should be accessible to most graduate students. The advanced material goes well beyond the literature; nearly one-third of the book is new. This work is indispensable for any topologist whose work includes four dimensions. It is a valuable reference for geometers and physicists who need an awareness of the topological side of the field.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them i...

- Brand: Brand: Princeton Univ Pr
- ASIN: 0691085773

###### 4-Manifolds (Oxford Graduate Texts in

Four Manifolds

This book presents the topology of smooth 4-manifolds in an intuitive self-contained way, developed over a number of years by Professor Akbulut. The text is aimed at graduate students and focuses on the teaching and learning of the subject, giving a direct approach to constructions and theorems which are supplemented by exercises to help the reader work through the details not covered in the proofs. The book contains a hundred colour illustrations to demonstrate the ideas rather than providing long-winded and potentially unclear explanations. Key results have been selected that relate to the material discussed and the author has provided examples of how to analyse them with the techniques developed in earlier chapters.

- ASIN: 0198784864 Part No: 9780198784869

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## Four Manifolds Video Results

### Handlebodies of Exotic 4-manifolds - Selman Akbulut

Graduate Workshop on 4-Manifolds: August 18 – 22, 2014 http://scgp.stonybrook.edu/archives/9733 Website of Selman Akbulut: http://selmanakbulut.com/ Related book: https://books.google.de/books/about/4_manifolds.html?id=1xMBDQAAQBAJ&source=kp_cover&redir_esc=y WIKIPEDIA: http://en.wikipedia.org...

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### Edward Witten - Monopoles & Four-Manifolds (1995)

Source: http://www.math.stonybrook.edu/Videos/Einstein/453-19950428-Witten.html Related paper: https://arxiv.org/abs/hep-th/9411102 WIKIPEDIA: https://en.wikipedia.org/wiki/Seiberg%E2%80%93Witten_invariants https://en.wikipedia.org/wiki/Donaldson_theory https://en.wikipedia.org/wiki/Edward_Witten

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### Jeffrey Meier: Bridge trisections of knotted surfaces in four-manifolds

Abstract: In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambie...

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