Monopoles and three-manifolds / Peter Kronheimer, Tomasz Mrowka.

Format:

Book

Author/Creator:

Kronheimer, P. B.

Contributor:

Mrowka, Tomasz.

Alumni and Friends Memorial Book Fund.

Series:

New mathematical monographs ; 10.

New mathematical monographs ; 10

Language:

English

Subjects (All):

Three-manifolds (Topology).

Homology theory.

Seiberg-Witten invariants.

Moduli theory.

Physical Description:

xii, 796 pages : illustrations ; 23 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2007.

Summary:

Originating with Andreas Floer in the 1980s, Floer homology has proved to be an effective tool in tackling many important problems in three- and four-dimensional geometry, and topology. This book provides a comprehensive treatment of Floer homology, based on the Seiberg-Witten monopole equations. After first providing an overview of the results, the authors develop the analytic properties of the Seiberg-Witten equations, assuming only a basic grounding in differential geometry and analysis. The Floer groups of a general three-manifold are then defined, and their properties studied in detail. Two final chapters are devoted to the calculation of Floer groups, and to applications of the theory in topology.

Suitable for beginning graduate students and researchers, this book provides the first full discussion of a central part of the study of the topology of manifolds since the mid 1990s.

Contents:

1 Monopole invariants of four-manifolds 2

2 Morse theory 14

3 Monopole Floer homology for three-manifolds 49

II The Seiberg-Witten equations and compactness 84

5 Compactness and properness 99

6 The blown-up configuration space 112

7 Unique continuation 120

8 Compactness in the blown-up configuration space 130

III Hilbert manifolds and perturbations 134

9 Completions and Hilbert manifolds 134

10 Abstract perturbations 152

11 Constructing tame perturbations 171

IV Moduli spaces and transversality 195

12 Transversality for the three-dimensional equations 196

13 Moduli spaces of trajectories 217

14 Local structure of moduli spaces 239

15 Transversality for moduli spaces of trajectories 265

V Compactness and gluing 274

16 Compactness of trajectory spaces 275

17 The moduli space on a finite cylinder 294

18 Stable manifolds and gluing near critical points 317

19 Gluing trajectories 343

VI Floer homology 375

20 Orienting moduli spaces 375

21 A version of Stokes' theorem 405

22 Floer homology 410

VII Cobordisms and invariance 449

23 Summary of results 449

24 The moduli space on a manifold with boundary 461

25 Maps from cobordisms 508

26 Composing cobordisms 535

27 Closed four-manifolds 551

28 Canonical gradings 581

VIII Non-exact perturbations 590

29 Closed two-forms as perturbations 590

30 Floer groups and non-exact perturbations 597

31 Some isomorphisms 605

32 Applications to gluing 622

33 Coupled Morse theory 634

34 Calculation of coupled homology 658

35 Application to the Floer groups HM 678

36 The manifold S[superscript 1] x S[superscript 2] 695

37 The three-torus 699

38 Elliptic surfaces 711

X Further developments 721

39 Homology spheres and negative-definite cobordisms 722

40 Genus bounds and scalar curvature 733

41 Foliations and non-vanishing theorems 741

42 Surgery and exact triangles 757.

Notes:

Includes bibliographical references (pages 779-784) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

9780521880220

052188022X

OCLC:

183149754

Online:

Contributor biographical information

Publisher description