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Smooth 4-Manifolds and the Seiberg–Witten Equations Contents Smooth 4-manifolds and the Seiberg–Witten equations T. PERUTZ UNIVERSITY OF TEXAS AT AUSTIN,SPRING SEMESTER, 2018 Contents 1 Classification problems in differential topology8 1.1 Smooth manifolds and their classification........................8 1.2 Dimensions 2 and 3....................................8 1.3 Higher dimensions: limitations..............................9 1.4 Higher dimensions: revised goals.............................9 1.5 4-manifolds........................................ 10 2 Review: The algebraic topology of manifolds 12 2.1 Cup products....................................... 12 2.1.1 Cechˇ cohomology................................ 12 2.2 De Rham cohomology.................................. 13 2.3 Poincare´ duality...................................... 13 2.3.1 The fundamental homology class........................ 13 2.3.2 Cap product and Poincare´ duality........................ 14 2.3.3 Intersection of submanifolds........................... 14 3 The intersection form 16 4 The cup product in middle dimensional cohomology 16 4.1 Symmetric forms over R ................................. 17 4.2 Characteristic vectors................................... 18 4.3 The E8 lattice....................................... 19 4.4 Topological examples................................... 20 5 The intersection form and characteristic classes 21 5.1 Cohomology of 4-manifolds............................... 21 5.2 Characteristic classes................................... 21 5.2.1 Stiefel–Whitney classes............................. 21 5.2.2 Chern classes................................... 23 5.2.3 Pontryagin classes................................ 25 2 T. Perutz 6 Tangent bundles of 4-manifolds 27 6.1 Vector bundles and obstruction theory.......................... 27 6.2 Obstruction theory.................................... 27 6.2.1 The Stiefel–Whitney classes as primary obstructions.............. 28 6.3 Vector bundles over a 4-manifold............................. 28 7 Rokhlin’s theorem and homotopy theory 30 7.1 Rokhlin’s theorem..................................... 30 7.2 The Pontryagin–Thom construction........................... 30 7.3 The J-homomorphism.................................. 31 7.4 Framed 3-manifolds and their bounding 4-manifolds.................. 32 7.5 Background: Relative homotopy groups......................... 33 7.6 Appendix: Effect on π3 of Lie group homomorphisms................. 33 8 Realization of unimodular forms by 4-manifolds 36 8.1 The theorem of Whitehead on 4-dimensional homotopy types.............. 36 8.2 Topological 4-manifolds................................. 37 8.3 Realizing unimodular forms by smooth 4-manifolds................... 38 8 Self-duality: linear-algebraic aspects 42 8.1 Self-duality........................................ 42 8.1.1 The Hodge star.................................. 42 8.1.2 2-forms in 4 dimensions............................. 42 8.1.3 Equivalence of conformal structures with maximal positive-definite subspaces. 43 8.1.4 Conformal structures as maps Λ− ! Λ+ .................... 44 8.2 2-planes.......................................... 45 9 Self-duality in 4 dimensions: Hodge-theoretic aspects 46 9.1 The Hodge theorem.................................... 46 9.1.1 The co-differential................................ 46 9.1.2 The co-differential as formal adjoint....................... 46 9.1.3 Harmonic forms................................. 46 9.1.4 Variational characterization............................ 47 9.1.5 The theorem................................... 47 9.2 Self-dual and anti-self-dual harmonic forms....................... 48 9.3 The self-duality complex................................. 48 9.4 The derivative of the period map............................. 49 Smooth 4-manifolds and the Seiberg–Witten equations 3 10 Covariant derivatives 51 10.1 Covariant derivatives in vector bundles.......................... 51 10.2 Curvature......................................... 52 10.3 Gauge transformations.................................. 54 10.4 Flat connections...................................... 54 10.5 Flat connections are local systems............................ 55 11 U(1)-connections 56 11.1 Connections and gauge transformations in line bundles................. 56 11.1.1 Chern–Weil theory................................ 56 11.1.2 Structure of BL .................................. 57 11.2 U(1)-instantons...................................... 58 12 Instantons in U(1)-bundles 59 12.1 Instantons in U(1)-bundles................................ 59 12.1.1 Generic non-existence.............................. 60 13 Differential operators 62 13.1 First-order differential operators............................. 62 13.2 Higher-order operators.................................. 64 13.3 Examples of symbols................................... 64 13.4 Elliptic operators..................................... 65 14 Analysis of elliptic operators 66 14.1 Fredholm operators.................................... 66 14.2 Sobolev spaces and elliptic estimates........................... 67 14.2.1 Sobolev spaces.................................. 67 14.3 Elliptic estimates..................................... 69 14.4 Lp bounds......................................... 70 15 Clifford algebras, spinors and spin groups 71 15.1 Clifford algebras..................................... 71 15.1.1 Orthogonal sums................................. 72 15.2 Spinors.......................................... 72 15.2.1 A quick note on spinors in odd dimension.................... 74 15.3 Projective actions..................................... 74 15.3.1 Projective action of the orthogonal group.................... 74 15.3.2 Projective action of the orthogonal Lie algebra................. 75 15.4 Spin groups........................................ 76 15.4.1 Clifford groups.................................. 76 15.4.2 Spin groups.................................... 77 15.4.3 Representations of spin.............................. 77 4 T. Perutz 16 Spin groups and spin structures in low dimensions 78 16.1 The compact Lie groups Spin(n)............................. 78 16.1.1 The story so far.................................. 78 16.1.2 Exponentials................................... 78 16.2 Spinors.......................................... 79 16.3 Low-dimensional cases.................................. 80 16.3.1 Spin(2)...................................... 80 16.3.2 Spin(3)...................................... 80 16.4 Spin(4).......................................... 81 17 Spin and Spinc -structures: topology 83 17.1 Spin structures on vector bundles............................. 83 17.1.1 Uniqueness.................................... 84 17.2 Existence and uniqueness in full............................. 84 17.3 Spinc -structures...................................... 85 17.4 Existence and uniqueness for Spinc -structures...................... 86 17.4.1 The case of 4-manifolds............................. 86 18 Dirac operators 88 18.1 The Levi-Civita connection................................ 88 18.2 Clifford connections................................... 88 18.3 The Dirac operator.................................... 90 18.4 The formal adjoint to a covariant derivative....................... 90 18.5 The Lichnerowicz´ formula................................ 91 19 The Seiberg–Witten equations 95 19.1 Spinc -structures in 4 dimensions............................. 95 19.2 Spinc -structures and self-duality............................. 95 19.3 The configuration space.................................. 96 19.4 The Seiberg–Witten equations.............................. 96 19.4.1 The Dirac equation................................ 96 19.4.2 The curvature equation.............................. 97 19.5 The index......................................... 98 19.5.1 A reinterpretation of the Seiberg–Witten index................. 99 20 The Seiberg–Witten equations: bounds 101 20.0.2 An inequality for Laplacians........................... 101 20.1 A priori bounds for solutions to the SW equations.................... 102 20.2 Finiteness......................................... 103 Smooth 4-manifolds and the Seiberg–Witten equations 5 21 The compactness theorem 105 21.1 Statement of the theorem................................. 105 21.2 Sobolev multiplication.................................. 105 21.3 The Seiberg–Witten equations in Sobolev spaces.................... 106 21.4 Elliptic estimates and the proof of compactness..................... 107 21.4.1 The positive feedback loop............................ 107 2 21.5 L1 bounds......................................... 108 2 2 21.5.1 From L1 to L3 .................................. 109 2 21.5.2 Weak and strong L1 convergence........................ 110 22 Transversality 112 22.1 Reducible solutions.................................... 112 22.1.1 Reducible configurations............................. 112 22.1.2 Reducible solutions and abelian instantons................... 112 22.2 Transversality for irreducible solutions.......................... 113 23 Transversality, continued 115 23.1 Generic transversality for irreducibles.......................... 115 23.1.1 Previously.................................... 115 23.1.2 The schematic argument............................. 115 23.2 Fredholm maps and the Sard–Smale theorem...................... 117 24 The diagonalization theorem 120 24.1 Statement and a preliminary reduction.......................... 120 24.2 Seiberg–Witten moduli spaces.............................
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