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Monopole Maps and L² -cohomology

By Tsuyoshi Kato

Monopole Maps and L² -cohomology Mathematics

Monopole Maps and L² -cohomology

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Description
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.
Product details
Number of Pages:
312
Release Date:
2025-08-26
Publication Date:
2025-08-26
Publisher:
Springer
Languages:
Original: English
ISBN10:
9819684714
ISBN13:
9789819684717
GPSR Manufacturer Reference:
[email protected]
Weight:
633 g
Height:
160 cm
Width:
241 cm
Thickness:
23 cm
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