The theme of this book is to establish a link between gauge theory and L ²-cohomology theory. Although both theories focus on differential topology, they have been developed rather independently. One of the main reasons lies in the differing characteristics of these theories. This book introduces an integrated theory that bridges these subjects. One goal of the book is to propose differential-topological conjectures that are covering versions of the so-called 10/8-theorem. We include various pieces of evidence to support them. This book is almost self-contained and is accessible not only to graduate students in differential geometry but also to both the experts in L ²-cohomology theory and gauge theory. This unique and fundamental book contains numerous unsolved problems, suggesting future directions of topology of smooth 4-manifolds by using various analytic methods.

After the introduction (Chap. 1), Chap. 2 gives a quick overview of the historical progress of differential topology. Chap. 3 covers the basic subjects of spin geometry. Chaps 4 and 5 deal with the foundations of the SeibergWitten and the BauerFuruta theories. In Chaps 6 and 7, we present the basic theory of L ²-cohomology, L ²-Betti numbers, amenability, and residual finiteness of discrete groups.

In Chap. 8, we treat the Singer conjecture and describe the solution to the conjecture for Kähler hyperbolic manifolds. We then describe various variations of Furuta's 10/8-inequalities and how the aspherical 10/8-inequalities conjecture is induced. We provide the evidence by examining various classes of 4-manifolds, such as aspherical surface bundles and complex surfaces.

Is the first book on SeibergWitten and L²-cohomology theories in a unified way and accessible to nonexperts Is the first book that explains about BauerFuruta theory in SeibergWitten theory Proposes new conjectures that arise from fusion of these theories and includes various evidence

Autorentext

Tsuyoshi Kato received his Ph.D. from Kyoto University in 1995. He is currently a Professor of Mathematics at Kyoto University.

Nobuhiro Nakamura received his Ph.D. from Kyoto University in 2006. He is currently a Professor of Mathematics at Fukushima Medical University.

Klappentext

The theme of this book is to establish a link between gauge theory and L²-cohomology theory. Although both theories focus on differential topology, they have been developed rather independently. One of the main reasons lies in the differing characteristics of these theories. This book introduces an integrated theory that bridges these subjects. One goal of the book is to propose differential-topological conjectures that are covering versions of the so-called 10/8-theorem. We include various pieces of evidence to support them. This book is almost self-contained and is accessible not only to graduate students in differential geometry but also to both the experts in L²-cohomology theory and gauge theory. This unique and fundamental book contains numerous unsolved problems, suggesting future directions of topology of smooth 4-manifolds by using various analytic methods. After the introduction (Chap. 1), Chap. 2 gives a quick overview of the historical progress of differential topology. Chap. 3 covers the basic subjects of spin geometry. Chaps 4 and 5 deal with the foundations of the Seiberg–Witten and the Bauer–Furuta theories. In Chaps 6 and 7, we present the basic theory of L²-cohomology, L²-Betti numbers, amenability, and residual finiteness of discrete groups. In Chap. 8, we treat the Singer conjecture and describe the solution to the conjecture for Kähler hyperbolic manifolds. We then describe various variations of Furuta's 10/8-inequalities and how the aspherical 10/8-inequalities conjecture is induced. We provide the evidence by examining various classes of 4-manifolds, such as aspherical surface bundles and complex surfaces.

Inhalt

Chapter 1 Introduction.- Chapter 2 A glimpse of progress of dierential topology.- Chapter 3 Spin geometry.- Chapter 4 SeibergWitten theory.- Chapter 5 BauerFuruta theory.- Chapter 6 𝑳^𝒑 cohomology.- Chapter 7 𝑳^2-Betti number and von Neumann trace.- Chapter 8 Aspherical 10/8 -inequality and Singer's conjecture.- Solutions.- References.- Index.