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Math 283 - Instanton Floer Homology

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Math 283 - Instanton Floer Homology Math 283 - Instanton Floer Homology Taught by Peter B. Kronheimer Notes by Dongryul Kim Spring 2018 The course was taught by Peter Kronheimer, on Mondays, Wednesdays, and Fridays from 11am to 12pm. There were three assignements throughout the course. Contents 1 January 22, 2018 5 1.1 Topology of smooth 4-manifolds . .5 1.2 Connections . .7 2 January 24, 2018 8 2.1 Curvature . .8 3 January 26, 2018 11 3.1 Moduli spaces of connections . 11 3.2 Elliptic operators . 11 3.3 Linear gauge-fixing . 12 4 January 29, 2018 14 4.1 Sobolev embeddings . 14 4.2 Moduli space of anti-self-dual connections . 15 5 January 31, 2018 17 5.1 Reducibles on M ........................... 18 6 February 2, 2018 19 6.1 Uhlenbeck's compactness theorem . 19 7 February 5, 2018 22 7.1 Uhlenbeck compactification . 22 7.2 Three-manifolds and the Chern{Simons functional . 23 1 Last Update: August 27, 2018 8 February 7, 2018 25 8.1 Chern{Simons functional under gauge transformations . 25 8.2 Representation variety . 26 9 February 9, 2018 28 9.1 Gradient flows and PDE . 28 10 February 12, 2018 30 10.1 SO(3)-bundle as a SU(2)-bundle . 30 10.2 Morse theory . 31 11 February 14, 2018 33 11.1 Operator on a cylinder . 33 12 February 16, 2018 35 12.1 Spectral flow . 35 12.2 Negative gradient flow . 35 13 February 21, 2018 37 13.1 Morse homology . 37 13.2 Morse theory on the space of connections . 38 14 February 23, 2018 40 14.1 Instanton Floer homology . 40 14.2 Uhlenbeck compactification of the space of flow lines . 41 15 February 26, 2018 43 15.1 Proof of compactification . 43 16 February 28, 2018 45 16.1 Good space of perturbations . 45 16.2 Holonomy perturbation . 46 17 March 2, 2018 48 17.1 Critical points of a holonomy perturbation . 48 17.2 Checking the bigness criterion . 49 18 March 5, 2018 51 18.1 Morse{Smale condition of perturbations . 51 18.2 Comparing Floer homology . 52 19 March 7, 2018 54 19.1 Cobordism . 54 20 March 9, 2018 56 20.1 Compactification of flowlines . 56 2 21 March 19, 2018 59 21.1 Marked bundles . 59 22 March 21, 2018 61 22.1 Marking on a torus . 61 23 March 23, 2018 63 23.1 Orbifolds . 63 24 March 26, 2018 65 24.1 Examples of bifolds . 65 24.2 Analysis on bifolds . 66 25 March 28, 2018 68 25.1 Fundamental group of a bifold . 68 25.2 Anti-self-duality equation on an orbifold . 69 26 March 30, 2018 70 26.1 Index between points . 70 27 April 2, 2018 72 27.1 Anti-self-duality on connected sum . 72 27.2 Instanton homology on bifolds is a functor . 73 28 April 4, 2018 75 28.1 Atomic bifolds (with markings) . 75 28.2 Grading of instanton Floer homology . 76 29 April 6, 2018 77 29.1 Operators on J #(K)......................... 77 30 April 11, 2018 79 30.1 Computation of the operators . 79 30.2 Nonvanishing theorem for webs . 80 31 April 13, 2018 81 31.1 Relations between operators . 81 32 April 16, 2018 83 32.1 More relations . 83 33 April 18, 2018 85 33.1 Excision . 85 34 April 20, 2018 87 34.1 Deformations of J # ......................... 87 3 35 April 23, 2018 88 35.1 Properties of homology with local coefficients . 88 36 April 25, 2018 89 36.1 Instanton Floer homology with coloring . 89 4 Math 283 Notes 5 1 January 22, 2018 The three main topics we will cover are: • 4-manifolds • Gauge theory (Donaldson's work in the 1980s) • 3-manifolds, knots, trivalent spatial graphs 1.1 Topology of smooth 4-manifolds Before Donaldson's work, nothing was know about 4-manifolds. Here, we mean smooth oriented closed 4-manifolds. The π1(X) can be any finitely generated group. We also have ∼ 2 ∼ H2(X; Z) = H (X; Z) = fcomplex line bundles over Xg via the Chern class c1(L). This isomorphisms also can be thought this way. Given a line bundle L and a section s, the vanishing locus s−1(0) is going to be an oriented 2-dimensional submanifold. Its homology class will be in H2. Proposition 1.1. Every σ 2 H2(X; Z) is represented by a 2-dimensional sub- manifold Σ2 ,! X4. The group H2 carries a quadratic form, the intersection product, and this is equivalent to H2 with the cup-square 2 2 4 H × H ! H (X; Z) =∼ Z: 2 This is symmetric and unimodular on H =torsion. In H2, this is the count of intersection points, with suitable orientation: X [Σ1][Σ2] = (±1): Σ1\Σ2 In particular, [Σ][Σ] is the degree of the normal bundle, i.e., the Euler class. This can be seen by deforming Σ to Σ0 along a section of the normal bundle ν. Because this is a symmetric unimodular form, we can consider its rank and signature: + − + − b2 = b + b ; σ = b − b : But unimodular indefinite forms over the integers is not classified by its signa- ture. Unimodular indefinite forms are either • odd forms λ(1) ⊕ µ(−1) or 0 1 • even forms λ( 1 0 ) ⊕ µ(±E8). Even forms have the property that x · x is always even. So the fundamental question we ask are: Math 283 Notes 6 • Which quadratic forms arise from 4-manifolds? • Does the quadratic form determine the 4-manifold if π1(X) = 0? Here are some examples of 4-manifolds. • X = S4: Here Q = 0. • X = S2 × S2: Here, the two S2 intersect once, and each have trivial 0 1 normal bundle. So Q = ( 1 0 ). • X = CP 2: The rank is 1 and Q = (1). • X = CP 2, with the opposite orientation: Here, Q = (−1). Now note that taking connecting sums of manifolds give the direct sum of in- tersection forms. This gives us all the odd forms. To get E8, we need something more interesting. Take X a K3-surface, and we are going to look at its tangent bundle. But let's take a step back. Given a 4 2 4 general X , we can consider w2(X) 2 H (X; Z=2), p1(X) 2 H , e 2 H4. In the π1 = 0 case, we have w2(X) ` y = y ` y: So if H2(X; Z) is torsion-free, every y 2 H2(X; Z=2) has an integer lift. So w2(X) = 0 if and only if [Σ][Σ] is even for all Σ (i.e., when the quadratic form is even). Also, we have 4 X i p1(X)[X] = 3σ; e(X)[X] = (−1) bi(X): 0 by the Hirzebruch signature theorem. Going back to the K3-surface, we see that c1(TX) = 0 and c2(TX) = 2 e(X) = 24. From w2 = c1 (mod 24) and p1 = c1 − c2, we obtain rank Q = b2 = 22. Also, σ = −8. This shows that 0 1 QX = 3( 1 0 ) ⊕ 2(−E8): 0 1 Unsolved question. Which of λ( 1 0 ) ⊕ µ(−E8) arise as QX ? In particular, 3 is always λ ≥ 2 µ? It can be shown that µ is even. If w2 = 0 then σ is divisible by 16. For a smooth complex surface, you can blow up at a point. This is taking the connected sum with CP 2. So we define 0 X = X#CP 2 2 as the blow-up in general. In particular, X = K3#CP has rank b2 = 23 and signature σ = −17. This is now odd, and QX = 3(1) ⊕ 20(−1): Going back to our uniqueness question, we can ask if K3#CP 2 =∼ 3CP 2#20CP 2. The answer is no, and the deep reason comes from gauge theory. Math 283 Notes 7 1.2 Connections Now we are going to study gauge theory. The main set up is • a n-manifold X, • a vector bundle E ! X, • and a connection A on E. This connection is a map 0 1 1 r :ΩX (E) ! ΩX (E) = Γ(ΛX ⊗ E) satisfying r(fs) = fr(s)+(df)⊗s. Given two vector bundles E; E0 and a map u : E0 ! E, we can pull back the connection A on E, and get a connection u∗(A) on E0. From now, I will consider objects (E; A), a bundle together with a connection. Math 283 Notes 8 2 January 24, 2018 I'm not talking about bundles and connections because I want to start from scratch, but because I want to make sure we're talking about the same things. Let G be a Lie group and g be the left-invariant vector fields. Let π : P ! X be a principal bundle so that G acts on P on the right. We have a subbundle V ⊆ TP given by V = ker π∗. Definition 2.1. A connection is a subbundle H ⊆ TP such that V ⊕H = TP at every point and is G-invariant. So given any path γ on X, there is a unique horizontal lift to P . This is parallel transport. We also have H as the kernel θ : TP ! V =∼ g · P; so where θ is a g-valued 1-form on P . Consider the associated vector bundle E ! X, coming from a finite-dimensional representation of G. Then we get a map 0 1 r :ΩX (E) ! ΩX (E): This is another way to think about connections. A representation is a map g ! End(RN ), so we get a map gP ! End(E); where gP is the associated vector bundle with fiber g.
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