18.937

Tom Mrowka

MIT

Fall Semester, 2011

Tom Mrowka 18.937 What is special about differential topology in dimension 4?

The basic tool in the high dimensional differential topology is Smale’s h-cobordism theorem. Theorem (Smale, 1962) Let X be a simply connected manifold of dimension n ≥ 6. ` Suppose that ∂X = M− M+ and that the inclusion M− ,→ X is a homotopy equivalence. Then

X is diffeomorphic to [0, 1] × M−.

In particular M− is diffeomorphic to M+.

Tom Mrowka 18.937 The basic tool in the proof is the Whitney Trick. This was introduced by by Whitney in 1943 in his proof of the strong Whitney embedding theorem. Theorem (Whitney, 1943) Let X be a smooth manifold of dimension n. Then X embeds in Rn. The trick is to remove double points in pairs by finding a Whitney disk. In manifolds of dimension 5 and higher we can find embedded Whitney disks by transversality arguments.

Tom Mrowka 18.937 In dimension 4 these arguments fail. Michael Freedman building on work of Casson and Bing showed that non the less one can find topological local flat Whitney disks in dimension 4. The basic problem that arises is understanding the number of double points of immersed S2’s or resolving the double points the genus of embedded surfaces.

Tom Mrowka 18.937 Regarding classification of four-manifolds we have Theorem (Milnor, 1958) Simply connected four-manifolds are homotopy equivalent and if only if their intersection forms are isomorphic. Wall proved the following. Theorem (Wall 1964) Simply connected homotopy equivalent four-manifolds are h-cobordant. In fact homotopy equivalent simply connected four-manifolds become diffeomorphic after connected sum with sufficiently many S2 × S2’s

Tom Mrowka 18.937 In 1986 Donaldson proved that the h-cobordism theorem definitely fails in dimension 4. He showed Theorem (Donaldson,1986) The manifold CP2]9(−CP2) admits 2 non-diffeomorphic smooth structures. Shortly after Robert Friedman and John Morgan showed Theorem (Friedman ,1988) The manifold CP2]9(−CP2) admits infinitely many non-diffeomorphic smooth structures.

Tom Mrowka 18.937 These last results where proved by early versions of Donaldson’ s invariant. These are invariants cooked up from the moduli spaces of Anti-Self-Dual connections on four-manifolds. These eventually take the form of an analytic function

DX : H∗(X) → R.

The basic theorems Donaldson proved about these invariants are Theorem (Donaldson, 1986) + + Suppose b (X) ≥ 3 and X = X1]X2 with b (Xi ) > 0. Then 2 DX ≡ 0. Connected sum with −CP does not kill the invariants. and Theorem (Donaldson, 1986) If X is smooth structure underlying a complex surface with + b (X) ≥ 3 then DX is not identically zero.

Tom Mrowka 18.937 Rochlin’s thereom and the 11/8-ths conjecture.

In the early 1950’s Rochlin prove the following theorem. Theorem (Rochlin) Let X be a spin 4-manifold then

σ(X) ≡ 0 (mod 16)

Tom Mrowka 18.937 In the early 1980’s Harer, Kas and Kirby amongst other tired to get a hands on the K 3-surface hoping possibly to show that one could split off an S2 × S2-summand. Below is a handle picture of a K3 surface (this one is due to Akbulut and Kirby.) Doesn’t look like it splits off such a summand.

Tom Mrowka 18.937 Are complex manifolds everything interesting in dimension 4?

Tom Mrowka 18.937 It turns out that DX can expressed in terms of which can be expressed in terms of finitely cohomology classes 2 Ki ∈ H (X; Z)/Tor and the intersection form Q. These classes constrain the genus of the embedded surfaces by the adjunction inequality valid for surfaces representing homology classes of non-negative self-interserction

S · S + χ(S) ≤ Ki · S.

Tom Mrowka 18.937 Milnor’s unknotting number conjecture for torus knots. This inequality has two natural predecessor’s. For a complex curve C ⊂ X where X is a complex surface we have the adjunction equality

C · C + χ(C) ≤ KX · C.

∗ 2 where KX = c1(T X) ∈ H (X, Z) is the canonical bundle. On the other hand if F is taut foliation of a three manifold Thurston proved χ(S) ≤ e(F) · S.

Tom Mrowka 18.937 Exotic smooth structures on R4.

Tom Mrowka 18.937 In 1986 Floer introduced his Instanton Floer homology for homology 3-spheres. I∗(Y ).

I∗(Y ) is the Morse homology of Chern-Simons functional and indeed there is a natural perfect pairing

I∗(Y ) ⊗ I∗(−Y ) → R.

Donaldson realized that I∗(Y ) was useful for computations of DX .

Tom Mrowka 18.937 2 If X = X− ∪Y X+ and Y is a homology sphere and b+(X±) ≥ 1 and given h ∈ H∗(X) written as h = h− + h+ we can find

DX± (h±) ∈ I∗(±Y ) so that

DX (h) = hDX− (h−), DX+ (h+)i These observations where motivations for Segal’s axioms for conformal field theories and Atiyah axioms of topological field theories. Note that there are strong restrictions above that do not appear in Atiyah’s definition.

Tom Mrowka 18.937 Early results on Donaldon’s invariants

Tom Mrowka 18.937 X a compact riemannian manifold X possibly with boundary. G ⊂ Aut(V) a compact Lie group. g ⊂ End(V), Lie algebra of G.

π : P → X a principal G bundle. E = P ×G V denote the associated V bundle. The vertical tangent space VTP = ker dπ. There is a canonical isomorphism of ιp : VTpP → g. adP = P ×ad g ⊂ End(E) and AdP = P ×Ad G ⊂ Aut(E).

Tom Mrowka 18.937 Forms with values in adP are a super lie algebra.

[· ∧ ·]:Λi (T ∗X) ⊗ adP ⊗ Λj (T ∗X) ⊗ adP → Λi+j (T ∗X) ⊗ adP so that [a ∧ b] = (−1)|a||b|+1[b ∧ a] Here if we write X a = ai ⊗ ξi and X b = bj ⊗ ηj where the ai and bj are forms and ξi and ηj are sections of adP we have X [a ∧ b] = ai ∧ bj [ξi , ηj ]. i,j

Tom Mrowka 18.937 In particular for one forms [a ∧ a] is not necessarily zero. These satisfy the Super-Jacobi Identity

[a∧[b∧c]]+(−1)|a|(|b|+|c|[b∧[c∧a]]+(−1)|c|(|a|+|c|[c∧[a∧b]] = 0

Now for a one form we have

[a ∧ [a ∧ a]] = 0.

Tom Mrowka 18.937 A connection A in P can be viewed many ways. A system of parallel transport in P

A G invariant subbundle HA ⊂ TP that is tranversal to the vertical tangent space VTP = ker(dπ) A g valued one form A on P which is ad-equivariant and is −1 ιp on the vertical tangent space. A covariant derivative

∞ ∞ ∗ dA : C (X; adP) → C (X, T X ⊗ adP).

Tom Mrowka 18.937 Observe that bundle automorphisms g : P → P a.k.a. gauge transformations are equivalent to maps gˆ : P → G so that

gˆ(ph) = h−1gˆ(p)h via the identifcation g(p) = pgˆ(p) then g(ph) = phgˆ(ph) = phh−1gˆ(p)h = pg(p)h. Thus gauge transformations are sections of AdP.

Tom Mrowka 18.937 The curvature of a connection measures the failure of HA to be h involutive. Given a vector X ∈ TpP let X denote the projection to HA|p. Then expression

˜ h ˜ h −A([X , Y ])|p ˜ ˜ measures the failure of HA to be involutive. Here X and Y are any extensions of X and Y to vector fields on P. We claim that this expression only depends on X and Y . If we multiply X˜ by a function f . Then ˜ ˜ h ˜ h ˜ h ˜ h ˜ h −A([hA(f X), Y )|p = −f (p)A([X , Y ]) + vA(Y f X ) = −f (p)A([X˜ h, Y˜ h]).

Tom Mrowka 18.937 Now we claim as a two form on P 1 F = dA + [A ∧ A]. A 2

Proof. Suppose that ξˆ = ιp(ξ), ηˆ = ιp(η) are vertical vectors. Then 1 0 = dA(ξ,ˆ ηˆ) + [A ∧ A](ξ,ˆ ηˆ) 2 = ξˆ(η) − ηˆ(ξ) − A([ξ,ˆ ηˆ]) + [A(ξˆ), A(ˆη)] = 0 − 0 − [ξ, η] + [ξ, η].

If ξˆ = ιp(ξ) and Y is horizontal we have

1 0 = dA(X, Y ) + [A, A](X, Y ) 2 = ξˆ(0) − Y (ξ) − A([ξ,ˆ Y˜ ]) + 0 = 0.

Tom Mrowka 18.937 If X and Y are horizontal then 1 dA(X, Y ) + [A, A](X, Y ) 2 = XA(Y ) − YA(X) − A([X˜ , Y˜ ]) + [A, A](X, Y ) = −A([X˜ , Y˜ ]).

As required.  Now if g is a gauge transformation then g∗A is another connection on P and we have

Fg∗(A) = Adgˆ ◦ FA.

∞ 2 Thus we can view FA ∈ C (X, Λ ⊗ adP).

Tom Mrowka 18.937 Often it is convenient to work in a trivialization, i.e. pull everything back by a local section

s : U → P.

Writing a = s∗A , a lie algebra valued one form we have 1 s∗(F ) = da + [a ∧ a]. A 2 The covariant derivative associated to A can be viewed as follows. Given e a section of E → X view e : P → V which is equivariant with respect to some representions ρ : G → End(V ). Then de is a one-forms with values in V . We can restrict this one form to HA and in this was we get a basic form

dAe = de +ρ ˙ ◦ A(e).

Tom Mrowka 18.937 Pulling this back to a local trivialization and specializing to the case of the adjoint representation we have

∗ s (dAe) = de + [a, e].

The set of (Ci nfty)-connections is a affine space for

Ω1(X; adP) since the difference of two connections is a basic one-form. Now write A0 = A + a. Then we wish to compute the curvature of A0. 1 F 0 = d(A + a) + [A + a ∧ A + a] A 2 1 = F + da + [A ∧ a] + [a ∧ a] A 2 1 = F + d a + [a ∧ a]. A A 2

Tom Mrowka 18.937 The curvature satisfies the Bianchi Identity

0 = dAFA

= dFA + [A ∧ FA] 1 = d(dA + [A ∧ A]) + [A ∧ F ] 2 A 1 1 = [dA ∧ A] − [A ∧ dA] + [A ∧ F ] 2 2 A = −[A ∧ dA] + [A ∧ FA] 1 = [A ∧ [A ∧ A]] = 0. 2 The last equality is consequence of the Jacobi-Identity.

Tom Mrowka 18.937 The Yang-Mills functional. The Yang-Mills energy of A is Z Z 2 E(A) = − tr(FA ∧ ∗FA) = |FA| ∗ 1. X X

(Recall the tr(A2) is a negative definite form on skew symmetric matrices.) The Euler-Lagrange equations for E is

∗ dAFA = 0 the Yang-Mills Equations. Recall that the curvature always satisfies the Bianchi Identity.

dAFA = 0

Informally A is a non-linear harmonic form!

Tom Mrowka 18.937 The Chern-Weil formula. Consider a homogeneous polynomial φ : g → R of degree k. We can also view P as multlinear map φ : gk → R. Suppose that P is ad-invariant.

φ(ξ1, . . . , ξk ) = P(Adgξ1,..., Adgξk ). Differentiating this equation gives also

0 = φ([ξ, ξ1], ξ2, . . . , ξk )+φ(ξ1, [ξ, ξ2], . . . , ξk )+. . . φ(ξ1, ξ2,..., [ξ, ξk ]).

Then given a connection we can construct a differential form

cφ(A) = φ(FA,..., FA).

Tom Mrowka 18.937 We claim that cP (A) is a closed form.

dcP (A) = kP(dFA, FA, FA,..., FA)

= kP(dFA + [A ∧ FA], FA,..., FA) (1) = 0

By the Bianchi Identity.

Thus cφ(A) defines deRham cohomology class. In fact the cohomology depends only on P and not the connection A. 0 To see this write A, A = A + a. Let At = A + ta. Then

t2 F = F + td a + [a ∧ a]. At A A 2 Then d F = d a + t[a ∧ a] = d a. dt At A At

Tom Mrowka 18.937 d d c (A ) = (F ,..., F ) dt φ t dt At At

= kcφ(dAt a, FAt ,..., FAt )

= k dcφ(a, FAt ,..., FAt ). Thus we have Z 1 0 cφ(A ) − cφ(A) = d(k cφ(a, FAt ,..., FAt ))dt 0 In particular notice that the difference is canonically exact, the primitive is called the Chern-Simons functional. Thus we get a characteristic class for P:

2k cφ(P) ∈ H (X).

Tom Mrowka 18.937 The Chern classes of complex vector bundle E → X(or Un-principal bundle) fit into this framework as follows. If we take write ∞ X k det(1 + tξ) = t φk (ξ) i=0 then i c (E) = [φ ( F )] ∈ H2k (X) k k 2π A

If we have a real vector bundle (or On-principal bundle) F → X we define k pk (F) = (−1) c2k (F ⊗ C)

Tom Mrowka 18.937 Notice that for an SU2-bundle we have

 ia b + ic F = A −b + ic −ia

Then −1 c (A) = (a ∧ a + b ∧ b + c ∧ c) 2 4π2 On the other hand −a ∧ a − b ∧ b − c ∧ c  F ∧ F = A A ...... −a ∧ a − b ∧ b − c ∧ c and tr(FA ∧ FA) = −2(a ∧ a + b ∧ b + c ∧ c) So we can also write 1 c (A) = tr(F ∧ F ). 2 8π2 A A

Tom Mrowka 18.937 If G = SO3 then we define the instanton number: 1 1 Z 1 = − h ( ), [ ]i = − ( ∧ ) ∈ k p1 P X 2 tr FA FA Z 4 8π X 4

Note if P lifts to an SU2 bundle Q then adQ ≡ adP and so

p1(P) = p1(adP) = p1(adQ) = −c2(adQ ⊗ C) = −c2(End(E)) = −4c2(E) where E denotes the C2-bundle associated to Q. and similarly if P lifts to a U2 bundle Q

2 p1(P) = −4c2(Q) + c1 (Q).

P is determined upto isomorphism by p1(P) and 2 w = w2(P) ∈ H (X, Z2).

Tom Mrowka 18.937 X an oriented riemannian n-manifold. The Hodge Star

∗ :Λk → Λn−k

Behavior under conformal change:

g 7→ e2σg =⇒ ∗ 7→ e(n−2k)σ∗

In particular if n = 2k Hodge star is conformally invariant. Also note that ∗2 = (−1)k(n−k) In particular for n = 4, k = 2

∗ :Λ2 → Λ2 has ∗2 = 1.

Tom Mrowka 18.937 Thus 2-forms have a conformally invariant decomposition.

Λ2 = Λ+ ⊕ Λ− ∗ = 1 ∗ = −1

Explicitly: e1, e2, e3, e4 oriented orthonormal frame.

1 2 3 4 1 3 2 4 1 4 2 3 ωI = e ∧e +e ∧e , ωJ = e ∧e −e ∧e , ωK = e ∧e +e ∧e span Λ+, the Self-Dual two forms and

e1 ∧ e2 − e3 ∧ e4, e1 ∧ e3 + e2 ∧ e4, e1 ∧ e4 − e2 ∧ e3 span Λ−, the Anti-Self-Dual two forms. Note that Λ+ and Λ− are pointwise orthogonal under both the riemmanian inner product and the wedge product.

Tom Mrowka 18.937 Thus we can decompose the curvature of a connection:

+ − FA = FA + FA .

+ − FA = 0 (FA = 0) are the Anti-Self-Dual (Self-Dual) Yang-Mills equation. Note that the Bianchi identity implies

dAFA = 0 and hence if A is SD or ASD we have

∗ dAFA = − ∗ dA ∗ FA = ± ∗ dAFA = 0.

Thus SD and ASD are critical points for E but more is true!

Tom Mrowka 18.937 If X is a closed four-manifold recall: Z Z 2 8π k = tr(FA ∧ FA), E(A) = − tr(FA ∧ ∗FA) ≥ 0. X X Thus adding these formulae gives: Z Z 2 − − E(A)+8π k = tr(FA∧(−∗FA+FA)) = −2 tr(FA ∧∗FA ) ≥ 0 X X while subtracting gives Z 2 + + E(A) − 8π k = −2 tr(FA ∧ ∗FA ) ≥ 0 X Thus E(A) ≥ 8π2|k| + − with equality if and only if FA = 0 when k ≥ 0 and FA = 0 when k ≤ 0.

Tom Mrowka 18.937 The basic instanton. We’ll use quaternionic notation. H = R + RI + RJ + RK where IJ = K + cyclic and IJ = −JI + cyclic. x = x0 + x1I + x2J + x3K Conjugation x = x0 − x1I − x2J − x3K .

7 2 S ⊂ H the unit sphere

Two different SU2 = Sp(1) = {x ∈ H|x¯x = 1} actions.

(x, y)q = (xq, yq) or (x, y)q = (qx¯ , qy¯ ).

In either case the quotient is S4 = HP1.

Tom Mrowka 18.937 Thus we have two principal bundles with total space S7.

4 P± → S where P+ has the right action and P− has the left action. These 4 are the unit sphere bundles of H-bundles S± → S . For example ¯ S+ = {([x, y], (v0, v1))|(v0, v1) = h(x0, x1), h ∈ H}.

Tom Mrowka 18.937 Give P± the connection A± which declares that the horizontal space at p ∈ P± is the orthogonal complement of the fiber 4 7 P± → S . Sp2(= Spin5) acts on S isometrically preserving the connections A±. The stabilizer Stabp of the point 7 2 p = (0, 1) ∈ S ⊂ H is a copy of Sp1(= SU2 = Spin3) of matrices of the form ∗ 0 0 0

The curvature of the connection is an SU2 equivariant map 2 ∗ 4 + − ∗ 4 Λ (Tp S ) = Λ ⊕ Λ (Tp S ) → adP±|p. At (0, 1) ∈ S7 we can write

7 T(1,0)S = H ⊕ imH.

The horizontal space is H ⊕ 0 and the vertical tangent space at p is identified with 0 ⊕ imH. Stabp acts trivially on the vertical tangent space and by left multiplication on P+ and right multiplication on P−.

Tom Mrowka 18.937 We need to understand the SU2 action. Write dx = dx0 + dx1I + dx2J + dx3K . Note that dx ∧ dx¯ is a purely imaginary 2-form (im H-valued). Indeed

dx ∧ dx¯ = −2(ωII + ωJ J + ωK K ).

In particular dx ∧ dx¯ is self-dual.

dgxh¯ ∧ dgxh¯ = gdx¯ ∧ dxg¯ .

Thus dx ∧ dx¯ is invariant under the left action ( and equivariant under the right action) so the left action is trivial on self-dual forms. Since Thus A+ is a self-dual connection. Similarly dx¯ ∧ dx is anti-self-dual and invariant under the left action and equivariant under the right action. Thus A− is an anti-self-dual connection.

Tom Mrowka 18.937 7 2 The connection A+ at the point (x, y) ∈ S ⊂ H is

Im(x¯ ∧ dx + y¯ ∧ dy).

We can trivialize P+ (or P−) by the section (x, 1) x 7→ 1 (1 + |x|2) 2

Show that the connection one-form Imx¯ ∧ dx a = (1 + |x|2) x¯ ∧ dx − x ∧ dx¯ = 2(1 + |x|2) represents A+ in a suitable trivialization of P+.

Tom Mrowka 18.937 We can construct from A+ other ASD-connections using conformal invariance. The dilatation τλ(x) = λx induces a conformal diffeomorphism. Indeed the basic instanton is 4 invariant under SO5 acting on S but the conformal group SO5,1 acts so effectively there is a

5 SO5,1/SO5 = H worth of ASD-connections in P+. Atiyah-Hitchin-Singer prove any ASD connections in P+ is gauge equivalent to one of these.

Tom Mrowka 18.937 This example exhibits the phenomenon of bubbling. As the the parameter λ 7→ ∞ the connections

Imλx¯ ∧ dλx τ ∗(a) = λ (1 + |λx|2) Imx¯ ∧ dx = (1/λ2 + |x|2) converge away from the origin to

Imx¯ ∧ dx (|x|2) which is gauge equivalent to zero! by the gauge transformation

g(x) = x/|x|.

Tom Mrowka 18.937 Spaces of connections and gauge transformations. A a connection. A is the space of C∞-connections. A is an affine space for the space Ω1(X; adP) = C∞(X; T ∗X ⊗ adP). g : P → P is an automorphism (gauge transformation) of P. g is a section of the bundle AdP. G the space of C∞-sections of AdP or group of gauge transformations. ∇Af or dAf The induced covariant derivative in any associated bundle, F = P ×ρ V . The action of g ∈ G. −1 g · A = A + gdAg . −1 −1 dAg means that covariant derivative of g thought of as a section of End(E) = E ∗ ⊗ E induced by A. Locally (g, a) 7→ gag−1 + gdg−1.

Tom Mrowka 18.937 We are interested in the studying connections up to isomorphism i.e. B = A/G. We need to understand how bad the action is.

Lemma g · A = A if and only if g is a parallel section of End(E). Proof. −1 −1 g · A = A =⇒ gdAg = −(dAg)g = 0. and hence dAg = 0. 

Tom Mrowka 18.937 1 N.B When G = SO3 the we can form the bundle Ad P of ”determinant 1” gauge transformations

1 Ad P = P ×SO3 SU2. where SO3 = SU2/Z (SU2) acts on SU2 by conjugation. The sections of this bundle G1 maps to G. Exercise 1 The map G → G has kernel Z2 if X is connected. The cokernel 1 isomorphic to H (X; Z2). We will often want to work with A/G1 rather than A/G. To deal with later we’ll choose submanifold Poincare´ dual to w2(P).

Tom Mrowka 18.937 Given a loop γ based at xo ∈ X the holonomy of A along γ the automorphism of P|xo given by parallel transport along γ.

holA(γ) ∈ AdP|xo .

The holonomy group of A at xo

holA(xo) = {holA(γ)|γ ∈ Ωxo (X)}

StabA is a subgroup of G and is the commutant of the holonomy group of A. Note that the center of G, Z (G) ⊂ StabA for all A. Definition

A is called reducible if StabA 6= Z (G).

Tom Mrowka 18.937 Exercise

Show that the possible subgroups of SO3 that appear are 1, Z2, Z2 × Z2, SO2, O2 and SO3 while if we restrict to the determinant 1 gauge group, the possible subgroups of SU2 that appear as Stab(A) for some connection are Z2, U1 and SU2. The quotient space B = A/G1 will have singularities if A contains reducible connections. Compare

n so3/SO3 where SO3 acts by the adjoint action. For each point in the quotient a neighborhood is modeled on a neighborhood of zero in one of the following three spaces. n−1 so3 . 2 n−1 n (R ) /SO2 × R . n The origin which has no better model than so3/SO3. We would like to show that B has similar local models.

Tom Mrowka 18.937 We will need to work with Sobolev spaces and Fredholm p n operators on them. There are the very basic facts. Let Lk (T ) be the completion of space of C∞ functions with respect to the norm. k Z p n X (j) p 1 n kf k p (T ) = |∇ f | dx ∧ ... dx . Lk n j=0 T We also will use Sobolev norms with fractional derivatives. These can be defined by interpolation. The L2-version can be easily defined by Fourier transform. If X f (x) = ˆf (n)eib·x n Then we 2 X 2 s/2 ˆ 2 kf k 2 = (1 + |n| ) |f (n)| Ls n

Tom Mrowka 18.937 Then we have the following properties.

p n q n Lk (T ) ,→ Ll (T ) If k − n/p ≥ l − n/q and k ≥ l. If both inequalities are strict the embedding is compact.

p n l,α n Lk (T ) ,→ C (T ) if k − n/p ≥ l + α where k ≥ l ≥ 0 and α > 0. If both inequalities are strict the embedding is compact.

Tom Mrowka 18.937 Furthermore when k − n/p and l − n/q are both negative

p n q n r n Lk (T ) × Ll (T ) ,→ Lm(T ) is continuous if

k − n/p + l − n/q ≥ r − n/m and k, l ≥ r.

q n Suppose that k − n/p is positive and k, l ≥ r then Ll (T ) is p n module over Lk (T ).

Tom Mrowka 18.937 In particular for n = 4 we have

2 4 L1 ,→ L , 2 4 p L2 ,→ L1 ,→ L for all p < ∞ 2 0 L26 ,→ C .

The following multiplications are continuous.

2 2 2 L1 × L1 7→ L , 2 2 p 2 L2 × L1 7→ L1 or Ls where p < 2 or s < 1.

Tom Mrowka 18.937 1 Note that ∇ ln(ln(r)) = r ln(r)

Z 1/2 1 Z 1/2 1 dr 1 ( )4 3 = ( ) = − −3( / ) < ∞. r dr 4 ln 1 2 0 r ln(r) 0 ln(r) r 4

4 4 0 showing that ln(ln(r)) ∈ L1(B1/2) but certainly not C .

Tom Mrowka 18.937 We need to complete our spaces of connections into Sobolev m m spaces. Suppose G = Um or SOm and let E be and C or R bundle so that P is the frame bundle of E. Using this connection there is a preferred gauge-equivariant Sobolev norm Z k p X i p kf k p = |∇Af | ∗g 1 . Lk,A X i=0 Any easy consequence of the Sobolev multiplication theorems p p is that the Lk,A-norm and the Lk,A0 are equivalent norms provided that difference is in 0 q ∗ A − A ∈ Ll,A(X, T X ⊗ adP)) p q p and the multiplication Lk × Ll → Lk−1 is continuous, i.e. l ≥ n/q − 1 and l > k − 1 and indeed there is a constant C > 0 0 depending on kA − A k q so that Ll,A

kekLp ≤ CkekLp . (2) k,A0 k,A

Tom Mrowka 18.937 Fractional order Sobolev norms. Fix a smooth connection A0 and consider the corresponding Laplace operator

∇∗ ∇ + 1 : C∞(X; adP) → C∞(X; adP) . A0 A0 This operator has compact inverse and one can construct compact operators

(∇∗ ∇ + 1)−s : Lp(X; adP) → Lp(X; adP) A0 A0 for any s > 0. For smooth A0 these operators are represented by a kernel with pole of order dist(x, y)−n+2s along the diagonal. p The space of Ls -sections of a bundle is

{(∇∗ ∇ + 1)−s/2e|e ∈ Lp(X; adP)}. A0 A0

This space has a norm using the connection A0,

∗ −s/2 k(∇ ∇ + 1) ek p = kek p (3) A0 A0 L L s,A0

Tom Mrowka 18.937 0 0 2 For any pair of connections A, A with A − A0, A − A0 ∈ Ls0 where s0 ≥ n/q − 1 and s0 > s − 1 the above definition of the 2 2 spaces L 0, and L 0, 0 and the corresponding norms k · kL2 s A s A s0,A and k · kL2 still make sense. Using that the fractional order s0,A0 spaces are interpolation spaces for the integral norms we see that the estimate (2) holds. When dependence of the norms on the choice of connection is not important for the discussion we will drop the connection from the notation for the norm. Note that if X is a manifold with boundary we can also define ´p Ls,A to be completion of smooth sections with support in the interior of X in the norm k · k p . Then we have the following Ls,A useful duality result for 1 ≤ p < ∞

´p ∗ q (Ls,A) = L−s,A.

Tom Mrowka 18.937 p p For consider the space Ak of Lk connections. The gauge group that acts naturally on this space of connections is

p p Gk+1 = {g ∈ Lk+1(X, adP)|g ∈ AdP a.e.}. When k + 1 − n/p > 0 this consists of continuous sections. Lemma

For G = Um or SOm and k + 1 − n/p > 0 is a Banach Lie group Proof. Consider the map

p p m : Lk+1(X, End(E)) 7→ Lk+1(X, End(E)) given by m(A) = AA∗ p −1 Then Gk+1 = m (1). For k + 1 − n/p > 0 m is a smooth map p since Lk+1 is a Banach algebra. M is also easily seen to have surjective differential and the kernel of differential always admits a complement (exercise). Tom Mrowka 18.937 Lemma p p p For k + 1 − n/p > 0, The map Ak × Gk+1 → Ak is a smooth map of Banach manifolds. Proof: In a trivialization the map is given by

(a, g) 7→ gag−1 + gdg−1

p p By the assumptions Lk+1 is a Banach algebra and Lk is module p over Lk+1. The above map is a composition of a continuous linear maps and continuous multiplications so it is smooth. 

Tom Mrowka 18.937 The space of connections, being an affine space is alway a Banach manifold. Lemma p p p For k + 1 − n/p ≥ 0, Bk = Ak /Gk+1 is a Hausdorff topological space.

2 Proof: We proof this in the case that of L2 gauge transformations. The general case is similar. We must show 2 2 2 2 that {(A, g · A)|A ∈ A1, g ∈ G2 } ⊂ A1 × A1 is closed. Fix p Ao ∈ Ak

i→∞ (Ai , gi · Ai ) −→ (A, B)

Ai = Ao + ai , gi · Ai = Ao + bi . Then −1 −1 gi dAo gi + gi ai gi = bi ⇒

dAo gi = gi ai − bi gi . (4)

Tom Mrowka 18.937 2 The sequences ai ,bi converge in L1 to a and b respectively. We 2 need to show that A0 + a is L2 gauge equivalent to A0 + b. We 4 4 see that dAo gi bounded in L so gi is bounded in L1. Now differentiate Equation (4)

∇2 g = ∇ g a − b ∇ g + g ∇ a − ∇ b g A0 i A0 i i i A0 i i A0 i A0 i i

The right hand side is bounded in L2 for each is either L2 × L∞ 4 4 or L × L . Thus we can pass to subsequence where gi 2 2 converges L2-weakly to g and strongly in L . The strong convergence implies again after passing to a subsequence a.e. convergence and so gg∗ = 1 a.e. thus g is gauge transformation. We can take the limit in Equation (4) in Lr for r < 2 showing that g takes A + a to A + b as required. .

Tom Mrowka 18.937 Orbifold structure for A/G. For concreteness consider we’ll 2 restrict to the L case and consider the space, A3/2, of connections on a four-manifold X. The gauge group, G5/2. Slices. Take orthogonal complement of tangent space of gauge orbit. 0 TAG = dAΩ (X, adP).

Using d tr(σ ∧ ∗a) = tr(dAσ ∧ ∗a) + tr(σ ∧ dA ∗ a) we have Z Z Z ∗ − tr(dAσ ∧ ∗a) = tr(σ ∧ dA ∗ a) = − tr(σ ∧ ∗dAa). X X X Thus we want ∗ 0 = hdAσ, ai = hσ, dAai. 0 ∗ for all σ ∈ Ω (X; adP) so dAa = 0. A ∈ A3/2 we have:

2 ∗ Ss,A() = {A + a|a ∈ L3/2,A(X; T X ⊗ adP), ∗ d a = 0 and kak 2 < } A Ls,A

Tom Mrowka 18.937 To state the result in general we need to discuss how to deal with reducible connections. If A is reducible StabA, the stabilizer of A, is a Lie subgroup of SOm. StabA acts on SA and freely on G5/2 and it is straightforward to see that the quotient

(SA,3/2() × G5/2)/StabA is a smooth Hilbert manifold. The tangent space at the equivalence class of (A + a, 1) is identified with the quotient of

2 ∗ 2 {α|α ∈ L3/2,A, dAα = 0} × L5/2,A(X, adP) by the finite-dimensional (and hence closed) subspace

0 0 0 {(−[a, ξ ], ξ )|dAξ = 0}.

Tom Mrowka 18.937 Proposition

For all A ∈ A3/2 there is an  > 0 so that the map

m :(S3/2,A() × G5/2)/StabA → A3/2 −1 −1 m([A + a, g]) = A + gag − (dAg)g is a G5/2-equivariant diffeomorphism onto its image.

Tom Mrowka 18.937 The proof will use the following facts. The operator

p p ∗ dA : LA,s(X, adP) → LA,s−1(X, T X ⊗ End(E)) has finite dimensional kernel and closed range. There is a closed ”Hodge” L2 orthogonal decomposition.

p ∗ p ∗ LA,s(X, adP) = ker(dA) ⊕ dA(LA,s+1(X, T X ⊗ adP)) A complement for the range of p p ∗ dA : LA,s(X, adP) → LA,s−1(X, T X ⊗ End(E)) is

∗ p ∗ p ker dA : LA,s−1(X, T X ⊗ adP) → LA,s−2(X, adP)

There is a constant λ1(A) > 0 so that for all ξ ⊥ ker(dA) we R 2 R 2 have X |dAξ| ≥ λ1 X |ξ| .

Tom Mrowka 18.937 Proof. 0 1 0 Write ξ = ξ + ξ for the Hodge decomposition where dAξ = 0 1 2 and ξ is L -orthogonal to elements of the kernel of dA. m is a local diffeomorphism if the differential of m is an isomorphism at (A, 1) Since m is G5/2-equivariant it suffices to check this at the equivalence classes of (A, 1). The differential is given by the map

D(A,1)m([α, ξ]) = dAξ + α

This map is surjective by the Hodge decomposition. If (α, ξ) is the kernel then α = 0 and ξ ∈ ker dA so [α, ξ] = 0.

Tom Mrowka 18.937 m is injective. It suffices to show that if g · (A + a) = A + b is in the slice then g ∈ StabA. The condition that g · (A + a) = A + b is equivalent to dAg = ga − bg. (5) ∗ Taking dA of this equation gives, using the slice condition ∗ ∗ dAa = dAb = 0,

∗ dAdAg = − ∗ dAg ∧ ∗a − ∗b ∧ ∗dAg. (6)

Use the Hodge decomposition g = g0 + g1 again and take the inner product of this equation with g1.

Tom Mrowka 18.937 1 2 1 1 kdAg kL2 ≤ (kakL4 + kbkL4 )kdAg kL2 kg kL4 1 1 ≤ κ(kakL4 + kbkL4 )kdAg kL2 kg k 2 L1,A −1 1/2 1 2 ≤ κ(1 + λ1 (A)) (kakL4 + kbkL4 )kdAg kL2 We have used

1 2 1 2 1 2 kg k 2 = kg kL2 + kdAg kL2 L1,A −1 1 2 ≤ (λ1(A) + 1)kdAg kL2 .

−1 1/2 1 So for κ(1 + λ1 (A)) < 1/2 we have u = 0 and hence 0 u = u is in Stab(A) as required. 

Tom Mrowka 18.937 Remark Not that the proof of injectivity of m only used L4-smallness. exercise Repeat this argument for D(1,A)m so show that for all A ∈ S1,A this differential is invertible. This then can be used to giva a sharper result that will be important in the proof of Uhlenbeck compactness.

Proposition (Big slices)

For all A ∈ A3/2 then the map

m :(S1,A() × G5/2/StabA) → A3/2 −1 −1 m([A + a, g]) = A + gag − (dAg)g is a G5/2-equivariant diffeomorphism onto its image. The image 2 contains an L1 ball about A.

Tom Mrowka 18.937 When we have a manifold with boundary the notion of slice needs to be modified. Z Z Z ∗ tr(dAσ ∧ ∗a) = tr(σ ∧ ∗a) + tr(σ ∧ ∗dAa). X ∂X X So the condition of being orthogonal to the tangent space of the gauge group implies formally

∗ dAa = 0 and ∗ a|∂X = 0.

Tom Mrowka 18.937 Elliptic boundary value problems from the Fourier transform view point. Recall that Z ˆf (ξ) = f (x)e−ihx,ξidx Rn is an isometry (upto factors of 2π)

2 n 2 n L (R , C) 7→ L (R , C) 1 kf k2 = kˆf k2 L2(Rn) (2π)n L2(Rn) Also the discrete Fourier transform Z ˆf (n) = f (x)e−ihx,nidx Tn and again

1 kf k2 = kˆf k2 L2(Tn) (2π)n `2(Zn)

Tom Mrowka 18.937 The Paley Wiener theorem gives a characterization of the Fourier transform of functions with support in (0, ∞). Theorem (Paley-Weiner) The Fourier transform is an isomorphism between

2 {f ∈ L (R; C)|sup(f ) ⊂ [0, ∞)} and the set of holomorphic functions H− so that h : H− toC so that there is a constant M so that for all η < 0 Z |h(ξ + iη)|2dξ ≤ M R

2 and so that h(ξ + iη) 7→L h(ξ).

Tom Mrowka 18.937 This is not so hard to prove. Note that if sup(f ) ⊂ [0, ∞) then Z ∞ ˆf (ξ + iη) = f (x)e−ixξ+ηx dx. 0 and ˆ 2 1 ηx kf (ξ + iη)kL2 = kf (x)e kL2 ξ 2π x

Tom Mrowka 18.937 Consider the typical boundary value problem for a first order operator for a Dirac type operator acting on section of a Clifford − ∞ bundle E → [∞) × Tn 1 D : C df f 7→ ( + Af , Pf (0)) dt where is a bundle map from E|0×Tn−1 → F. Taking the Fourier transform of the equation

df + Af = g dt gives iξˆf + f (0) + Aˆf = gˆ ˆ and we must have that f ∈ H−. Now we have

ˆf = (A + iξ)−1(gˆ − f (0)).

Tom Mrowka 18.937 In this lecture we prove Uhlenbeck’s Fundamental Lemma. Lemma Let B = be a geodesic ball in a riemannian manifold. Let Γ be the connection arising from a trivialization P = ×G for principal G bundle. There are positive constants C, o so that for all 2 ∗ R A = Γ + a with a ∈ L1(B, T B × g) and B tr(FA ∧ ∗FA) ≤ o there 2 is a map g ∈ L2(B, Mn(C)) with g ∈ G a.e. so that if we set b = gag−1 + gdg−1 we have ∗ d b = 0 and ∗b|∂B = 0 and R 2 2 R 2 B(|∇Γb| + |b| ) ∗ 1 ≤ C B |FA| ∗ 1.

Tom Mrowka 18.937 Proof. B = expx (B0(δ)). The first move in the proof is notice that we can replace the L2-norm of the curvature by an equivalent norm. We exploit this possibility by changing the metric to the pullback metric from the metric from Tx B. Let ∗˜ denote the Hodge star for this metric. The proof of this is a version of the continuity argument. Let Z 2 ∗ V = {a ∈ L 3 (B, T B ⊗ g)| − tr(FA ∧ ∗˜FA) ≤ }. 2 B

2 We use a density argument to get the L1-case. Let W ⊂ V be the subset where there is a C > 0 so that conclusions of the lemma holds.

Tom Mrowka 18.937 We will show that for  small enough and C large enough that:

V is connected.

W is closed.

W is open in V.

Thus W = V. V is connected. Let for 0 ≤ t ≤ 1 let τt : Bx (δ) → Bx (tδ) denote the scaling by t along geodesics through x. Consider the path of connection ∗ At = τt (A|Bx (tδ)). 2 We claim that for A an L3/2-connection this path is continuous 2 in the L3/2-topology. For t 6= 0 this is the same point that translation is point-wise continuous on Lp spaces. At t = 0 continuity follows directly from the definitions.

Tom Mrowka 18.937 Since the L2-norm of the curvature is conformally invariant and dilation is a conformal map we have that Z Z

− (tr(FAt ∧ ∗˜FAt )) = − (tr(FA ∧ ∗˜FA)) ≤  B Bx (tδ)

R ˜ and limt7→0 − B(tr(FAt ∧ ∗FAt )) = 0.

Tom Mrowka 18.937 W is closed in V. Let Ai = Γ + ai be a sequence in W which 2 converges in the L3/2-topology to A = Γ + a ∈ V. There is also then a sequence of gauge transformations gi so that gi · Ai = Γ + bi and the bi are in Coulomb gauge and satisfy the 2 estimates, so it follows that bi converge weakly in the L1 topology to b . Now consider the sequence of gauge transformations gi . They satisfy the equations:

dgi = gi ai − bi gi .

4 4 Thus dgi ∈ L and so gi ∈ L1 (gi is bounded). Taking the gradient of both sides gives:

∇dgi = ∇gi ai + gi ∇ai − ∇bi gi − bi ∇gi

2 hence gi is bounded in L2. Thus after passing to a 2 subsequence we may assume the gi converge weakly in L2 to g.

Tom Mrowka 18.937 subseq a.e |gi | ≤ M =⇒ gi −→ g.

Hence g ∈ Un almost everywhere.

4 4 4 2 subseq L L L ,L1 kai k 2 , kbi k 2 , kgi k 2 ≤ M =⇒ ai * a, bi * b, gi → g. L1 L1 L2

4− where ai , bi converge strongly in L and gi converges p 2 strongly in L for all p and in L1 so that in the equation

dgi = gi ai − bi gi . boths sides converges strongly in L2. Thus the we have

dg = ga − bg so that Γ + a is gauge equivalent to Γ + b. To see that Γ + a ∈ W we need to see that b satisfies the estimate and that 2 b ∈ L3/2

Tom Mrowka 18.937 Z Z 2 |FΓ+b| = hdb + b ∧ b, db + b ∧ bi U U Z ∗ 2 4 ≥ |db + d b| − kbkL4 U Z 2 2 ≥ |∇b| − CkbkL2 U 1 2 ≥ ((1 + λ1)/2 − C)kbk 2 . L1 We’ve used the identity Z Z Z Z ∗ 2 2 2 (|(d + d )b| = |∇b| + ∗b ∧ ∗db + |b| ∗∂U 1. U U ∂U ∂U

Tom Mrowka 18.937 2 ∗ Check that b ∈ L3/2. Take d of this equation giving

∆g = − ∗ (dg ∧ ∗a) + gd ∗a + ∗(∗b ∧ dg).

Rewrite as:

∆g −∗ (∗b ∧ dg)= − ∗ (dg ∧ ∗a) + gd ∗a.

Note that we also have ∗dg|∂B = ∗a|∂b. The red terms both are 2 2 2 2 in L1/2 since L3/2 × L1 7→ L1/2. Thinking of the blue term as function of g it is a bounded operator

2 2 L5/2 → L1/2 with operator norm controlled by kbk 2 . The map L1 h 7→ (∆h, ∗dh|∂U ) viewed a map

2 2 2 L5/2(U) → L1/2(U) × L1(∂U) has kernel and cokernel the constant sections.

Tom Mrowka 18.937 The equation has index 0 and notice that no matter what b is constant sections are in the kernel and the cokernel. In small perturbation the kernel can only jump up so b = 0 so for b with 2 2 small L1-norm it follows that g ∈ L5/2. The equation b = gag−1 − g−1dg 2 says that b is in L3/2.

Tom Mrowka 18.937 W is open in V Openness follows from the Big Slices Lemma, Lemma 9 which tells us that if A is gauge equivalent into the 2 SΓ,1() then so is a neighborhood of A in the L3/2-topology. 2 Finally suppose that Ai is a sequence of L3/2 connections 2 connections in V converging to an L1 connection A then following the proof of closedness we see that A can be put into a good gauge and this representative satisfies the estimates. 

Tom Mrowka 18.937 The same circle of ideas proves the following lemma which will later be used in the proof of the Uhlenbeck compactness theorem. Lemma (Curvature is Proper) 2 With o as in the previous lemma. Let SV(0/2) be the set of L1 connections A = Γ + a with Z 2 |FA| ∗ 1 ≤ 0/2 B and ∗ d a = 0 and ∗ a|∂B = 0. Then the curvature map

2 2 ∗ F• : SVo/2 → L (B, Λ (T B) ⊗ adP)

A → FA is proper.

Tom Mrowka 18.937 Proof. Suppose FAi is a Cauchy sequence. Z Z 2 2 |FAi −FAj | = |dai + ai ∧ ai − daj − aj ∧ aj | ∗ 1 B B Z 2 2 ≥ (|dai − daj | − |ai ∧ ai − aj ∧ aj | ) ∗ 1 B Z ∗ 2 2 ≥ |(d + d )(ai − aj )| − (kai kL4 + kaj kL4 )kai − aj kL4 B Z 2 2 ≥ |∇(ai − aj )| − 2Ckai − aj kL2 B 1 2 ≥ ((1 + λ)/2 − 2C)kai − aj k 2 . L1

Thus Ai is also Cauchy.

Tom Mrowka 18.937 An SO3 bundle on P → X has two characteristic classes

4 2 p1(P) ∈ H (X; Z) ≡ Z and w2(P) ∈ H (X; Z2)

These are constrained by

℘(w2(P)) ≡ p1(P)(mod 4). where 2 4 ℘ : H (X; Z2) → H (X; Z4) is the Pontraygin square.

Tom Mrowka 18.937 Given P → X and riemannian metric g let

+ Mk,w = {A ∈ A(P)|FA = 0}/G(P).

1 2 where k = − 4 hp1(P), [X]i and w = w2(P) ∈ H (X; Z2).

Tom Mrowka 18.937 Fredholm maps and Kuranshi description of the moduli space. Definition A smooth map of Hilbert manifolds f : M → N is called Fredholm if for all x ∈ M

dx f : Tx M → Tf (x)N is a Fredholm map.

Tom Mrowka 18.937 Theorem If f : M → N is a Fredholm map and x ∈ M then there is

a neighborhood 0 3 U ker(dx f ) ⊂ Tx M and neighborhood x 3 W ⊂ M.

a map κ : U → coker dx f . an immersion φ : U → M.

So that φ(0) = x, d0φ : T0U → ker(dx f ) ⊂ Tx M and so that

φ(κ−1(0)) = f −1(0) ∩ W.

Tom Mrowka 18.937 If A is an ASD connection then we have the deformation complex

d+ 2 dA 2 1 A 2 + 0 7→ L3(X; adP) → L2(X;Λ ⊗ adP) → L1(X;Λ ⊗ adP) → 0.

This is an elliptic complex and the index or of the complex is

3 −8k + (χ(X) + σ(X)) 2 We call the negative of the index

3 8k − (χ(X) + σ(X)) 2 the formal dimension of the moduli space.

Tom Mrowka 18.937 This index was computed by Atiyah-Hitchin-Singer using the index theorem. The result is. Theorem The map A + a 7→ F +(A) is a Fredholm map viewed as a map

2 + φ : SA,2 → L1(X, Λ (X)).

The index of φ is 3 8k − (χ(X) + σ(X)) + dim Stab(A) 2

Tom Mrowka 18.937 Examples. 4 4 5 1. X = S , k, M1,0(S ) = B and dimension formula gives 3 8k − (2 + 0) = 8k − 3 2 Note that k = 1 case checks with the example. 2.X arbitrary, k = 0. A is a trivial connection. The complex becomes 3 copies of

+ 2 d 2 1 d 2 + 0 7→ L3(X) → L2(X;Λ ) → L2(X;Λ ) → 0. where the dimensions of the cohomology groups are b0(X), b1(X), b+(X) respectively. 3 − (χ(X) + σ(X)) 2 3 = − (2b0(X) − 2b1(X) + b+(X) + b−(X) + b+(X) − b−(X)) 2 = −3(b0(X) − b1(X) + b+(X)).

Tom Mrowka 18.937 These calculations can be turned around to prove the dimension formula using excision.

Tom Mrowka 18.937 The Chern-Simons Function. Given A connection in P → X. Z 2 tr(FA ∧ FA)(mod 8π Z) X depends only on the gauge equivalence class of A|∂X . Indeed if A0 is a connection in P0 and there is a bundle iso 0 0 g : P|∂X → P∂X so that gA = A then Z Z Z 2 tr(FA ∧ FA) − tr(FA0 ∧ FA0 ) = tr(FA00 ∧ FA00 ) ∈ 8π Z. X X DX DX is the double of X and A00 is a connection in a bundle P” → DX obtained by patching together P with the connection A and P0 with the connection A0 using g.

Tom Mrowka 18.937 Definition Let B be a connection in Q → Y Z 2 CS(B) = tr(FA ∧ FA)(mod 8π Z) X

Another viewpoint on CS. Given a pair of connections B0 and B1 choose Bt , t ∈ [0, 1]. View this path as a connection A in [0, 1] × Q → [0, 1] × Y . Z

CSB0 (B1) = tr(FA ∧ FA). [0,1]×Y

If B0 comes from a trivialization and B1 = B0 + b and we set Bt = B0 + tb so that:

t2 t2 F = d(tb) + [b ∧ b] = dt ∧ b + tdb + [b ∧ b]. A 2 2

Tom Mrowka 18.937 Then 2 tr(FA ∧ FA) = dt ∧ tr(2b ∧ (tdb + t [b ∧ b])) and so Z Z 2
tr FA ∧ FA) = dt ∧ tr(2b ∧ tdb + t b ∧ [b ∧ b] [0,1]×Y [0,1]×Y Z 1 = tr(b ∧ db + b ∧ b ∧ b). Y 3

Exercise

If B0 does not arise from a trivialization show that Z 1 CSB0 (B1) = tr(2b ∧ FB0 + b ∧ db + b ∧ [b ∧ b]) Y 3

Tom Mrowka 18.937 Proposition

Suppose G = SU2. Then P = Y × SU2 and g can be viewed as a map g : Y → SU2. We claim

2 CS(B1) − CS(gB1) = 8π deg(g) Proof: In this case the difference is the Chern-Weil integral for a connection in the bunlde arising from the mapping torus of g. The c2(P) is the Euler class of the associated complex two plane bundle. There is a section of this mapping torus bundle with deg(g) transverse zeroes.

Tom Mrowka 18.937 The Chern-Simons functions provides another local characterization of Anti-Self-Duality. Proposition (Mean Value Property) Suppose is connected smooth manifold with boundary. A in → P X the Z − tr(FA ∧ ∗FA) ≥ −CS(A|∂X ) X with equality if and only if A is ASD. Proof: Z Z Z + + − tr(FA ∧ ∗FA) + tr(FA ∧ FA) = −2 tr(FA ∧ ∗FA ). X X



Tom Mrowka 18.937 Uhlenbeck’s compactness theorem. Let U be a small geodesic ball in a Riemannian four-manifold. Since all bundles on a ball are trivial we write M(U) for the moduli space on U. For  > 0 write Z  2 M (U) = {[A] ∈ M(U)| |FA| ∗ 1 ≤ }. U

Proposition Let U be a geodesic ball in a Riemannian four-manifold. There 0 is an o so that for any proper subball U ⊂ U the set of connections the restriction map

Mo (U) → M(U0)

0 2 is a compact map (Here we give M(U ) the L1 quotient topology)

Tom Mrowka 18.937 Proof: Take a sequence of gauge equivalence classes of connections [Ai ] We can choose the representatives Ai = Γ + ai so that they are in Coulomb gauge and we have uniform 2 L1-estimates on ai . Pass to a subsequence where ai converge 2 2 weakly in L1. By Fubini’s theorem for each i ai |∂Br is in L1 except for r in a set of measure zero. Since the sequence is 2 countable for all i, ai |∂Br ∈ L1 a.e in r. Now fix such an r. Since the sequence is uniformly bounded (indeed as small as we wish by readjusting 0) we can pass to a subsequence where ai |∂Br 2 is uniformly bounded by a small constant in L1. Now we can 2 pass to a subsequence where ai |∂Br converges strongly in L1/2.

Tom Mrowka 18.937 In particular

lim CS(Ai |∂Br ) = CS(A∞|∂Br ) i7→∞ and hence

lim kFA kL2(B ) = kFA∞ kL2(B ) i7→∞ i r r So the sequence of curvature converges in L2! It looks like we can apply the properness of curvature result applies but our connections one-forms no longer satisfy the boundary condition

∗ai |∂Br = 0.

Tom Mrowka 18.937 2 What we do know is that ∗ai |∂Br is L1 small. Now consider the map

∗ ξ −ξ ξ −ξ ξ −ξ ξ −ξ ξ 7→ (d (e ae + e de ), ∗(e ae + e de )|∂Br ) Z 2 {ξ ∈L5/2(Br , adP)| ξ ∗ 1 = 0} Br Z 2 2 3 → {ξ ∈ L1/2(Br , adP)| ξ ∗ 1 = 0} ⊕ {ξ ∈ L1(∂Br , Λ ⊗ adP)} Br 2 ∗ 2 This map for our strange ai ∈ L1 with d ai = 0 and ai |∂Br ∈ L1 and both with small norm this map is a well defined smooth map.

Tom Mrowka 18.937 Note that the terms involving a in the first component are

dg · ag−1 − ga · dg−1. and the Sobolev multiplication

2 2 2 L3/2 × L1 → L1/2 is continuous, while the boundary term is easily seen to be continuous for . The differential is an isomorphism when a = 0 and so for a small. Thus we can put the ai in Coulomb gauge 2 by an L5/2 gauge transformations. Note that this gauge transformations are all homotopic to the identity so they do not change the value of the Chern-Simons function.

Tom Mrowka 18.937 We can assume the gauge transformations converge strongly 2 0 ˜ 2 ˜ in L2 ∩ C in particular ai |∂Br still converges in L1/2 and ai still 2 ˜ ˜ converges weakly in L1 to a. Set A = Γ + a. Since Ai are ASD, the convergence of CS implies R |F |2 ∗ 1 converges to Br Ai R |F |2 ∗ 1. This implies strong convergence of F and hence Br A Ai by the properness of the curvature map the strong convergence to a˜i as required.

Tom Mrowka 18.937 We can also deduce regularity of solution from a similar discussion. Theorem 2 + Let A be an L1-connection. Suppose that FA = 0. Then there is 2 ∞ a an L2-gauge transformations g so that g · A is C . Having done this we can sharpen the above compactness theorem a compactness theorem in the C∞-topology. Proposition Let U be a geodesic ball in a Riemannian four-manifold. There 0 is an o so that for any proper subball U ⊂ U the set of connections the restriction map

Mo (U) → M(U0) is a compact map (Here we give M(U0) the C∞ quotient topology)

Tom Mrowka 18.937 Theorem Let G be a compact Lie group. Let P → X be principle G-bundle Let Ai be a sequence of ASD-connections with Z 2 |FAi | ≤ M X then after passing to a subsequence there are

a collection of finitely many points x1,..., xn ⊂ X, a bundle P0 → X. a sequence of bundle isomorphisms 0 gi : P|X\{x1,...,xn} → P |X\{x1,...,xn} , a connection A in a bundle P0 → X so that for any open subset U b X \{x1,..., xn} gi · Ai ∞ converges in the C -topology to A|X\{x1,...,xn}.

Tom Mrowka 18.937 Proof. We can pass to a subsequence where the sequence of 2 curvature densities |FAi | ∗ 1 converges in the weak topology to a density ω. If we cover X by balls of radius 2−N then for each N so that the balls of half the radius cover as well. We choose the covers in such a way that that any point of X is in at most K -balls of any of the given covers. Then for each N we can find i(N) so that at most 2MK /o of these ball have Z 2 |FAi | ≤ o/2 Bα for all i ≥ i(N).

Tom Mrowka 18.937 We can pass to a further subsequence so that the centers of these bad balls converge to points x1,... xn. We thus get an exhaustion of X \{x1,... xn} by a sequence of balls Bα where Z 2 |FAi | ≤ o/2. Bα Finally by a diagonalization argument and the compact inclusion lemma we can pass to a subsequence where there are sequence giα of gauge transformations to that giαAi |Bα ∞ converges to a connection Γ + aα on Bα in the C -topology.

Tom Mrowka 18.937 We need to patch these local connections and gauge transformations together. The sequence of transition functions

−1 ηi,αβ = giα|Bβ giβ |Bα satisfy the equation

dηi,αβ = ηi,αβai,β − ai,αηi,αβ.

In particular this means that the ηi,αβ are uniformly bounded in the C1-norm. Arzela-Ascoli then tells us that we can pass to a 0 subsequence where ηi,αβ where the ηi,αβ converge in the C topology. Then the equation tells us the the ηi,αβ converge in 1 the C -topology. Repeating this bootstrap tells us that ηi,αβ converges in the C∞-topology. 

Tom Mrowka 18.937 Removeable singularites. Theorem (Uhlenbeck) Let P → X be a principle bundle. Suppose that A is finite energy ASD connection on X \{x1,..., xn}. Then there is a 0 0 bundle P → X and a connection A in P so that A|X\{x1,...,xn} 0 and A |X\{x1,...,xn} are gauge equivalent.

Definition

A sequence of connection Ai in P → X converges upto bubbling to A ∈ P0 → X if there is a collection of points {x1,..., xn} and a sequence of gauge transformations gi so that for every compact subset K b X \{x1,..., xn} so that ∞ Ai |K 7→ A|K in the C -topology.

Tom Mrowka 18.937 What happens at the bubbling points. This requires a little more sophisticated discussion. Fix attention to one such point x ∈ X. Then there is a sequence of points {xi } converging to x and radii {ri } converging to zero so that Z 2 |FAi | ∗ 1 ≥ 0/2. Bxi (ri )

By increasing the ri we can assume that Z 2 |FAi | ∗ 1 ≥ 0/2. Bx (ri )

Further increasing the ri we can assume that Z 2 |FAi | ∗ 1 = 0/4. Bx (r)\Bx (ri )

Tom Mrowka 18.937 Using the exponential map pull the connections back to balls in the tangent space Tx X. Rescale the balls so that ri = 1. The sequence of connections converges on compact sets away from a finite set of points contained in the unit ball of Tx X to an ASD connection A with Z 2 |FA| ∗ 1 ≤ 0/4. Tx X\B1(0)

The connection A extends over the S4 so it has integral energy. At each of the points in the unit ball we can do the same rescaling. In the end we have accounted for all the energy that is lost at x.

Tom Mrowka 18.937 Together with the compactness theorem this leads to the Uhlenbeck compactification. Definition

The Uhlenbeck compactification UMk (X) of Mk (X) is the closure of Mk (X) ⊂ Mk (X)∪X ×Mk−1(X)∪Sym2X ×Mk−1(X)∪...∪Symk (X)×M0(X) with respect to the upto bubbling topology.

Remark The subset of Mk (X)∪X ×Mk−1(X)∪Sym2X ×Mk−2(X)∪...∪Symk (X)×M0(X) which actually appears in the closure is important to understand. Taubes’ gluing work showed how to understand this subset in many situations.

Tom Mrowka 18.937 Floer’s Instanton Homology

Let Y be a 3-manifold P = Y × SU2. Γ the trivial connection and B = Γ + b. Z 1 CS(B) = tr(b ∧ db + b ∧ [b ∧ b]) Y 3 Floer’s great insight was that using the analytic tools developed by Uhlenbeck, Taubes and Donaldson one could carry over the Morse complex construction to CS viewed as a function on B = A/G.

Tom Mrowka 18.937 d To compute the directional derivative ds CS(B + sc)|s=0 introduce the one parameter family of 4d-connections on [0, 1] × Y with t being the coordinate in [0, 1]

As = B + st c

Then s2t2 F = F + d (4)(stc) + [c ∧ c]. As B B 2 so that d F | = dt ∧ c + td c. ds As s=0 B

Tom Mrowka 18.937 Here is the computation of the derivative of Chern-Simons

d d Z CS(B + sc)|s=0 = tr(FAs ∧ FAs )|s=0 ds ds [0,1]×Y Z 1 Z = 2 tr((dt ∧ c + tdBc) ∧ FB) 0 Y Z 1 Z = 2 dt ∧ tr(c ∧ FB) 0 Y Z = 2 tr(c ∧ FB) Y

Thus critical points of CS are flat connections FB = 0.

Tom Mrowka 18.937 Given a riemannian metric on Y we can formally compute the gradient of CS and we have Z Z 2 tr(FB ∧ c) = 2 tr(∗FB ∧ ∗c) = −2h∗FB, ci. Y Y

1 The downward gradient flow for L = − 2 CS is dB 1 = −∇ L = ∇ CS = − ∗ F . dt B 2 B B

Tom Mrowka 18.937 A major miracle occurs here. If we view a solution B(t) as a connection A on R × Y the connection satisfies the Anti-Self-Dual Yang-Mills equation

FA = − ∗4 FA.

As a 4d-connection dB F = dt ∧ + F A dt B and dB ∗F = dt ∧ ∗ F + ∗ A 3 B 3 dt This later equations is invariant not just under G3 = {g : Y → SU2} but under G4 = {h : R × Y → SU2}.

Tom Mrowka 18.937 Problems.

gradient flow of CS not well defined Hessian of CS, ∗dB has infinitely many positive and negative eigenvalues. B is singular. CS is not single valued (cf Novikov homology). Compactification of moduli spaces (including gluing). Construct pertubations to achieve Morse-Smale Condition that do no destroy compactness properties.

Tom Mrowka 18.937 1 The gauge group G = {g : Y → SU2} acts on Ω (Y ) ⊗ su2 by

g · b = gbg−1 + gdg−1.

Under this action we have

−1 CS(g · B) = CS(B) + deg(g) and Fg·B = gFBg

Thus the set of critical points of CS is preserved by the G action and {a|FB = 0} = Hom(π1(Y ), SU2)/conj = R(Y ) where the identification is by the holonomy representation.

Tom Mrowka 18.937 Examples: 3 S . Only the trivial representation ρtriv . R(Y ) = pt. Stab(ρtriv ) = SU2.

Poincare sphere, SU2/Binary Icosohedral group. Has three representations upto conjugacy ρtriv and ρdef and ρ¯def 1 2 S × S . R(Y ) = SU2/conj an interval. 3 1 1 1 T . R(Y ) = S × S × S /{Z2}. (Fintushuel-Stern) S(p, q, r) a Seifert fibered homology sphere with three exceptional fibers. The representation space is a finite set of isolated points. (Fintushuel-Stern) S a Seifert fibered homology sphere with four exceptional fibers. The representation space is a union of S2’s

Tom Mrowka 18.937 Notice that a reducible representation into SU2 lives in a 1 U1-subgroup. We can understand Hom(π1(Y ), U1) = H (Y ; U1) via the long exact sequence

1 1 1 2 2 H (Y , Z) → H (Y ; R) → H (Y ; U1) → H (Y , Z) → H (Y , R)

Thus 1 b1 2 H (Y ; U1) ≡ U1 × Tor(H (Y , Z)). 1 Thus for a three manifold H (Y ; U1) = {0} if and only if H1(Y ; Z) = 0, i.e. Y is a homology sphere.

Tom Mrowka 18.937 Here is Fintushel and Stern’s example. Recall that S(p, q, r) is the intersection of

p q r 5 3 V (p, q, r) = x + y + z = 0 ∩ S ⊂ C .

This a Seifert fibered space with the S1-action

u · (x, y, z) = (uqr x, upr y, uqpz).

The quotient is an orbifold S2 with three orbifold points with orders p, q and r.

Tom Mrowka 18.937 For a global quotient like this we can form the homotopy quotient

(S(p, q, r) × ES1)/S1 = S(p, q, r)//S1

The fundamental group of S(p, q, r)//S1 is called the orbifold fundamental group and has a presentation in this case as follows.

1 p q r π1(S(p, q, r)//S ) ≡ {t, u, v|t = 1, u = 1, v = 1, tuv = 1} = Tp,q,r

Tom Mrowka 18.937 Furthermore the long exact homotopy sequence looks like

1 1 → π1(S ) → π1(S(p, q, r)) → π1(S(p, q, r)//S ) → 1. or → Z → π1(S(p, q, r)) → Tp,q,r → 1.

Thus π1(S(p, q, r)) has a presentation

{t, u, v, h|tp = h−b1 , uq = h−b2 , v r = h−b3 , tuv = hb} for some integers b, b1, b2, b3. The data (b, p/b1, q/b2, r/b3) are called the Seifert invariants.

Tom Mrowka 18.937 ˜ ˜ ˜ For ρ : π1(S(p, q, r)) → SU2 is a representation write T , U, V and H˜ for the images of the generators. Then H commutes with the others so if ρ is not trivial H˜ = ±1. Then the other matrices satisfy the equations

T˜ p = ±1, U˜ q = ±1, V˜ r = ±1, T˜U˜ V˜ = ±1.

In particular pushing the representation into SO3 they give rise to a representation of the triangle group Tp,q,r . These representation we can understand rather easily. Let T , U and V ˜ denote the images in SO3 of T etc. Then T , U, V are rotations about axes `T , `U and `V by angles 2πk/p, 2πl/q and 2πm/r. The fact that these three element satisfy TUV = 1 implies that the the products of these rotations is the identity.

Tom Mrowka 18.937 The distinction between SU2 representations and SO3 representations. Suppose that ρ1 and ρ2 are two SU2 representations which which project to the same SO3 representation. Then there is a map δ : π1 → Z2 so that

ρ2(γ) = δ(γ)ρ1(γ).

Indeed δ is a representation so if Y is a Z2 homology sphere ρ1 and ρ2 are the same. Suppose that ρ˜ is an SO3 representation. For each γ ∈ π1(Y ) choose r(γ) ∈ SU2 a lift. Now r may not define a representation. Rather

r(γ)r(γ0) = η(γ, γ0)r(γγ0) where η : π1(Y ) × π1(Y ) → Z2 is a Z2 group 2-cocycle. Again if 0 0 Y is a Z2-homology sphere η(γ, γ ) = λ(γ)λ(γ ) so that ρ(γ) = λ(γ)r(γ) is a representation.

Tom Mrowka 18.937 The distinction between SU2 gauge equivalence classes and SO3-gauge equivalence classes of connections. Suppose that A1 and A2 are two connections which are gauge equivalent as SO3 connections. Then we have an open cover {Uα}, transition functions g¯αβ : Uα ∩ Uβ → SO3 and local connections one forms 1 aα ∈ Ω (Uα; so3) satisfying ¯ ¯−1 ¯ ¯−1 aα = gαβdgαβ + gαβaβgαβ since su2 = so3 we have can lift the connection to an SU2 on each Uα without making any further choices. Thus lifting the connection is equivalent to lifting the bundle. Choose lifts of the transition functions gαβ : Uα ∩ Uβ → SU2. These may fail to define transition functions because

ηαβγ = gαβgβγgγα : Uα ∩ Uβ ∩ Uγ → Z2 maybe a non-trivial Cˆ ech 2-cocycle. If ηαβγ is cohomologous to zero by a Cˆ ech 1-cochain λαβ then we can modify the lifts to SU2 replacing gαβ by λTomαβg Mrowkaαβ we get18.937 transition functions for a bundle. In any case [ηαβγ] represents the the nd 2 -Steifel-Whitney class of the SO3 bundle. Given a pair of critical points α, β set Z M(α, β) = {A| − tr(FA ∧ ∗FA) < ∞, FA = − ∗ FA, Z lim [A|t×Y ] = α, lim [A|t×Y ] = β}/G4. t7→−∞ t7→∞

Tom Mrowka 18.937 Note that A is ASD if and only if Z − tr(FA ∧ ∗FA) = ” − CS(α) + CS(β)” Z where the difference is computed with respect the path B(t) = A|t×Y .

Tom Mrowka 18.937 Fix Q → Y an SO3 or SU2 bundle. Given B± flat connections with trivial stabilizer choose a path B(t) with t ∈ R so that for t < −1 B(t) = B− and t > 1 B(t) = B+. Let A be the corresponding 4d-connection on and R × Q = P → Z = R × Y . Then we make the following definitions.

2 ∗ A(α, β) = A + L2(Z ; T Z ⊗ adP)

2 ∗ G = g ∈ 1 + L3(Z, End(E))|gg = 1. Since gA is locally gag−1 + gag−1 G acts on A(α, β)

B(α, β) = A(α, β)/G.

Tom Mrowka 18.937 Tom Mrowka 18.937 Remark If stabilizer of end points is not Z(G) this is not a good definition.

Remark

The definition depends on a homotopy class γ ∈ π1(BQ; α, β) so write Aγ(α, β) or Bγ(α, β) when necessary.

Tom Mrowka 18.937 For A ∈ A(α, β)

+ 2 + FA ∈ L1(Z, Λ (Z) ⊗ adP) Set + Mγ(α, β) = {[A] ∈ B(α, β)|FA = 0}

We’d like to see that Mγ(α, β) has a Kuranishi local structure and that [ M(α, β) = Mγ(α, β)

γ∈π1(BQ ,α,β)

Tom Mrowka 18.937 Write A = B + b + cdt where B is a pull back connection. Then

1 F = F + dt ∧ b˙ + d b − dt ∧ d c + [b ∧ b] + dt ∧ [c ∧ b] A B B B 2 1 0 = F + = dt ∧ (b˙ + ∗F + d c + ∗d b + ∗ [b ∧ b] + [c ∧ b]) A B B B 2 1 + ∗b˙ + F + ∗d c + d b + [b ∧ b] + ∗[c ∧ b]. B B B 2 The slice condition

∗ 0 = −dB(b + cdt) = ∗4dB ∗4 (b + cdt) = ∗dBdt ∧ ∗b + ∗c

= ∗4(−dt ∧ (dB ∗ b + dt ∧ ∗c˙ + dB ∗ c) ∗ ∗ = c˙ + dBb + dt ∧ dBc.

Tom Mrowka 18.937 Linearized equations at B.  ˙      b ∗dB dB b 0 = + ∗ c˙ dB 0 c

d This of the form dt + D where D is a first order self-adjoint elliptic operator. Theorem If D is an invertible first order self-adjoint elliptic operator acting on a vector bundle F → Y then d + D : L2(Z; F) → L2 (Z, F) dt k k−1 is invertible.

Tom Mrowka 18.937 Sketch of proof in the case k = 1. Note that Z d 2 d d k( + D)ukL2(Z ) h( + D)u, ( + D)ui ∗ 1 dt Z dt dt Z du du = | |2 + |Du|2 + 2h , Dui ∗ 1 Z dt dt Z du d = | |2 + |Du|2 + hu, Dui ∗ 1 Z dt dt Z du = | |2 + |Du|2 ∗ 1 Z dt 2 ≥ Ckuk 2 . L1

d 2 2 Thus dt + D : Lk (Z ; F) → Lk−1(Z , F) is injective with closed range. Exercise: Show the range is dense. Generalize this to the case of general k. .

Tom Mrowka 18.937 This can be used to prove in general that provided the Extend Hessian of L   ∗dB dB EHB = ∗ dB 0 is invertible for B = B± the linearized ASD equations and gauge fixing are Fredholm so the package we use to analyze the moduli space on closed manifolds carries over to this case. EHB plays the role of the Hessian in finite dimensional (and no group action) Morse theory.

Tom Mrowka 18.937 What does invertibility mean? If

b c is in the kernel of EHB and B is flat the b is a harmonic 1 0 representative of H (Y , adB) and c ∈ H (Y , adB). In other words the B must be Irreducible Infinitesimally isolated (non-degenerate critical point.)

Tom Mrowka 18.937 Given A ∈ Aγ(α, β) write A = B + cdt where B = B(t) is path in AQ and we can ask what is the index of d + D . dt B

We now have a family of self-adjoint operators DB(t). Such a family has a spectral flow. Below we have a spectral flow of +2.

Tom Mrowka 18.937 Note the self-adjoint Fredholm operators SF has three components SF = SF− ∪ SFo ∪ SF+ where SF± are the essentially positive and negative operators. SF± are contractible while

ΩSFo ≡ Z × BO( or Z × BU for complex ops).

Theorem (Atiyah-Patodi-Singer..) d Ind( + D ) = sf (D ) dt t t

Tom Mrowka 18.937 Idea of proof. Index is homotopy invariant so homotope the family so that Dt all have the same eigenvectors. Then consider d T = + ± tanh(t): L2( ) → L2( ). ± dt 1 R R

The kernel of T+ is spanned by sech(t) while for T− the kernel ∗ is spanned by cosh(t). Since T± = −T∓ when

Ind(T±) = ±1 = sf (± tanh(t)).

.

Tom Mrowka 18.937 Note the spectral flow is the what remains of the Morse index. µ(α) should be the number of negative eigenvalue of the Hessian. Infinite in this case. However in the finite dimensional situation

dim(M(x, y)) = µ(y) − µ(x) = sf (Hessγ(t)f ) and we can make sense in this case of the difference

µ(β) − µ(α) = sf (EHγ(t))

Tom Mrowka 18.937 1 If γ : S → BQ is a closed loop i.e. or γ˜ :[0, 1] → AQ and so that there is a gauge transformation with gγ˜(0) =γ ˜(1) then sf may be non-zero! Indeed let A be the connection in S1 × Q with

3 sf (D (t)) = Ind(D ) = 8k − (0 + 0) = 8k = 8 deg(g). γ A 2

1 N.B. if Q is an SO3 bundle k ∈ 2 Z.

Tom Mrowka 18.937 Lemma Let A be a finite energy ASD connection on the half-cylinder [0, ∞) × Y . Suppose that all flat connections Γ on Y have H1(Y , adΓ) = 0. Then there is a flat connection Γ,T > 0 and gauge transformation g on [T , ∞) × Y so that

g · A = Γ + a

2 ∗ and a ∈ L2([T , ∞) × Y ; T ⊗ adP).

Tom Mrowka 18.937 The first step in the proof is Lemma

Let Ai be a sequence of ASD connection on a four manifold X R | |2 7→ with boundary. Suppose that X FAi 0. Then there is a flat connection Γ and a subsequence still call Ai and a sequence of gauge transformations gi so that

gi · Ai 7→ Γ in the C∞-topology on compact subsets of the interior of X. Proof. The assumption that the curvature has L2 tending to zero preclude bubbling so the rest follows from the version of Uhlenbeck’s theorem we have already proved.

Tom Mrowka 18.937 Near constant solutions on a cylinder. Let Γ be a flat connection. Suppose that Γ is non-degenerate in the in sense that H1(Y , adΓ) = 0. We can repeat the estimate in Uhlenbeck’s theorem to prove. Lemma There is a constant C > 0, and a gauge invariant 2 L -neighborhood UΓ ⊂ A 2 (I × Y ) of Γ so that for for any 1 L2 connection A ∈ U we have the estimate Z Z 2 2 2 |∇Γa| + |a| ≤ C |FA| I×Y I×Y

Tom Mrowka 18.937 Proof. Let

2 ∗ SΓ() = {Γ + a|a ∈ L , dΓ a = 0, ∗a|∂I×Y , kak 2 < } Γ,2 LΓ,1 be a slice. Then as in the proof of Uhlenbeck Fundamental Lemma we have that G 2 ·SΓ() L3 is open in A 2 (I × Y ) and U will be an open subset. L2

Tom Mrowka 18.937 For connections A = Γ + a ∈ SΓ() we have Z Z 2 |FA| = hdΓa + a ∧ a, dΓa + a ∧ ai I×Y U Z ∗ 2 4 ≥ |dΓa + dΓ a| ∗ 1 − kakL4 I×Y Z 2 2 2 ≥ C1 (|∇Γa| + |a| ) ∗ 1 − C2kakL2 I×Y Γ,1 2 ≥ (C1 − C2)kak 2 . LΓ,1

We’ve use the estimate Z Z ∗ 2 2 2 |dΓa + dΓ a| ∗ 1 ≥ C1 (|∇Γa| + |a| ) ∗ 1 I×Y I×Y for one forms with a ∗ |∂I×Y This follows from the fact that H1(I × Y ; adΓ) = H1(Y ; adΓ) = 0.

Tom Mrowka 18.937 We can combine this will elliptic bootstrapping to get Lemma For any k > 0 and any I0 a proper subinterval of I, there is a 2 constant C > 0, and a gauge invariant L1-neighborhood UΓ ⊂ A 2 (I × Y ) of Γ so that for for any connection A ∈ UΓ we L2 have the estimate

Z k Z X j 2 2 ( |∇Γa| ) ∗ 1 ≤ C |FA| 0 I ×Y j=0 I×Y

Tom Mrowka 18.937 For a gradient flow on a finite dimensional manifold there is a dichotomy for flow lines passing near a non-degenerate critical point. Either they approach the critical exponentially fast or the leave the neighborhood in finite time. Here is the proof whose outline we will follow in the infinite dimensional situation. Consider the a solution x(t) of

x˙ = −∇x f .

The main tool is the following differential inequality satisfied by f (t) ≡ f (x(t)). Lemma

Let xo be a non-degenerate critical point of f . Then there is a neighborhood U of xo and a constant µ > 0 so that df (x(t)) ≤ −µ(f (x(t)) − f (x )) dt o

Tom Mrowka 18.937 Proof. From the gradient flow condition we have

df = −|∇ (t)f |2. (7) dt x

Since xo is non-degenerate it has a neighborhood U so that for all y ∈ U we have

2 2 d(y, x) ≤ C1|∇x (t)f | . (8)

After possibly shrinking U further we can arrange that

2 f (y) − f (x0) ≤ C2d(y, x) (9) and hence that df ≤ −µf . dt We also have 2 f (y) − f (x0) ≤ µ|∇x (t)f | . (10)  Tom Mrowka 18.937 Corollary

If xo is a non-degenerate critical point then there is a neighborhood U so that for any flowline x(t) or the negative gradient of f contained in U we have

−µt f (x(t)) − f (x0) ≤ (f (x(0)) − f (xo))e ˙ Proof. We can assume f (xo) = 0. The inequality f ≤ −µf (x(t)) implies d (eµt f (x(t))) ≤ 0. dt Integrating this from 0 to t gives

eµt f (x(t)) − f (x(0)) ≤ 0 or f (x(t)) ≤ f (x(0))e−µt



Tom Mrowka 18.937 The above inequality says nothing if f (x(t)) ≤ f (xo) but changing f to −f and t → T − t we conclude

−µ(t−T ) −f (x(T − t)) + f (x0) ≤ (−f (x(T )) + f (xo))e

In particular x(t) is in contained in U on [−T , T ] we have

| − f (x(t)) + f (x0)| −µ(t+T ) −µ(T −t) ≤ min{−f (x(T ) + f (xo))e , f (xo) − f (T )e } ≤ C cosh(µt)e−µT

Tom Mrowka 18.937 We can repeat these are arguments to get the same results for the Chern-Simons function. We will prove the following Proposition For each non-degenerate flat connection Γ there is a gauge 2 invariant neighborhood UΓ in the L1-topology and a constant µΓ so that we have Z 2 CS(B) − CS(Γ) ≤ µΓ |FB| . (11) Y Note that Inequality (11) is the analogue of Inequality (10).

Tom Mrowka 18.937 The proof is similar to the finite dimensional case, and in fact we’ve already proved Lemma For any non-degenerate flat connection Γ there is a gauge invariant neighborhood UΓ and a constant C so that for all B ∈ UΓ there is a gauge transformation g so that

g · B = Γ + b

∗ is in Coulomb gauge dΓ b = 0 and we have Z Z 2 2 2 (|∇Γb| + |b| ) ∗ 1 ≤ C |FB| ∗ 1. Y Y

Tom Mrowka 18.937 Lemma For any non-degenerate flat connection Γ there is a gauge invariant neighborhood UΓ and a constant C so that for all B ∈ U we 0 2 CS(B) − CS(Γ) ≤ C d 2 ([B], [Γ]) L1 Proof. Put B into Coulomb gauge with respect to Γ.

Z 1 CS(Γ + b) − CS(Γ) = tr(b ∧ db + b ∧ b ∧ b) Y 3 2 3 ≤ C(kbk 2 + kbk 2 ) LΓ,1 LΓ,1 2 ≤ C(1 + )kbk 2 . LΓ,1

Tom Mrowka 18.937 Having proved these estimates we have the following Proposition If Γ is a non-degenerate flat connection then there is a neighborhood UΓ of Γ and constants C, µ > 0 so that for any Anti-Self-Dual connection A on [−T , T ] × Y so that A|t×Y ∈ UΓ for all t ∈ I. Then we have

−µT |CS(A|t × Y ) − CS(Γ)| ≤ Ce cosh(µt) Combing this with Proposition 23 we have that on any interval of fixed length say [t − 1, t + 1] × Y ⊂ [−T , T ] × Y an ASD connection is gauge equivalent to one of the form Γ + a where a satisfies:

Z k X j 2 −µT ( |∇Γa| ) ∗ 1 ≤ Ce cosh(µt). [t−1,t+1]×Y j=0

Tom Mrowka 18.937 Lets write R∗(Y ) = {[ρ] ∈ R(Y )|[ρ]isirreducible}. Recall that we ∗ have R(Y ) = [t] ∪ R (Y ) if an only Y is a Z-homology sphere where [t] is the trivial gauge equivalence class. since if ρ : π1(Y ) → SU2 is reducible then ρ(π1) ⊂ U1 ⊂ SU2 and ρ factors through a non-trivial representation

ρ˜ : H1(Y , Z) → U1.

Tom Mrowka 18.937 Assumption. Suppose that Y is a homology sphere and for all α, β ∈ R(Y ) and all γ ∈ π1(BQ; a, b)

Mγ(a, b) cut out transversally by the ASD equations. Note that if a or b ∗ are in R (Y ) then all [A] ∈ Mγ(a, b) are irreducible and we have Then we have dim(Mγ(a, b)) = sfγ(a, b). Let ˘ Mγ(a, b) = Mγ(a, b)/R. ˘ So that dim(Mγ(a, b)) = sfγ(a, b) − 1.

Tom Mrowka 18.937 What is the dimension of Mγ(a, t) or Mγ(t, b). Define sfγ(t, b) as follows

and sfγ(a, t) similarly.

Tom Mrowka 18.937 Then

dim(Mγ(a, t) = sfγ(a, t) and dim(Mγ(t, a) = sfγ(t, b) but

sfγ1 (a, t) + sfγ2 (t, b) + 3 = sfγ1+γ2 (a, b)!

Tom Mrowka 18.937 What is the Uhlenbeck compactification of this space? An ideal instanton, I is a pair consisting of a point in a symmetric product of Z and an ASD connection on Z . Set [ k IMγ(a, b) = Sym (Z) × Mγ−k (a, b). k for the space of Ideal instantons from a to b in the homotopy class γ. The dimension of a typical piece k Sym (Z ) × Mγ−k (a, b) is 4k + sfγ(a, b) − 8k = sfγ(a, b) − 4k The moduli space of ideal instantons mod translation in the homotopy class γ is ˘ IMγ(a, b) = IMγ(a, b)/IR 3 I˘ where now the typical piece has dimension

sfγ(a, b) − 4k − 1

Tom Mrowka 18.937 A broken path of ideal instantons from a to b in the homotopy ˘ ˘ ˘ ˘ class γ is an ordered set (I1,..., Il ) with Ii ∈ Mγi (ai−1, ai ) and so that

if ai−1 = ai and γi = 0 then k > 0.

a0 = a, al = b

γ1 + ... + γl = γ. Let ˘ + Mγ (a, b) be the set of broken path of ideal instantons.

Tom Mrowka 18.937 Note that the dimension of a stratum

k0 kl (Sym (Z)×Mγ0−k0 (a0, a1))/R×... (Sym (Z)×Mγl −kl (al−1, b))/R

˘ + of Mγ (a, b) is

sfγ(a, b) − l − 4(k0 + ... + kl ) − 3t where t is the number times the trivial connection appears amongst the a1,..., al−1. Theorem With the topology of convergence upto bubbling mod ˘ + translation. Mγ (a, b) is compact. NB. The gauge equivalence class of the trivial connection t can be among the ai .

Tom Mrowka 18.937 The (mod 2) Floer-Morse complex is M C∗ = Z2a a∈R∗(Y ) with X X ˘ ∂(a) = ]{[A] ∈ Mγ(a, b)}β. ∗ b∈R (Y ) sfγ (a,b)=1 The homology of this complex is denoted

I∗(Y ).

Why is this ∂2 = 0? The component of ∂2(a) along c is

X X ˘ ˘ ](Mγ1 (a, b))](Mγ2 (b, c)) b∈R∗(Y )

Tom Mrowka 18.937 Consider the compactification

˘ + Mγ (a, c) where in this case we have sfγ(a, c) = 2. Only the non-negative dimensional strata appear so

2 − l − 4(k1 + ... + kl ) − 3t ≥ 0.

˘ + Thus ki = 0, t = 0 and l = 1, 2. So the only ends of Mγ (a, c) are ˘ ˘ Mγ1 (a, b) ∪ Mγ2 (b, c). Since these appear as ends the of a one dimensional moduli space there is an even number.

Tom Mrowka 18.937 Functorality and Invariance. Let X be a cobordism from Y0 to Y1, where the Yi are homology spheres, we’ll assume also that 1 b (X) = 0. Let given a ∈ R(Y0) and b ∈ R(Y1) we can define a moduli space. Mγ(a, X, b) Here γ is labeling the components of B(a, X, b). The formal dimension of this moduli space will be denoted

sfγ(a, X, b) with the same convention as before for the trivial connection.

Tom Mrowka 18.937 We construct an analogous compactification from the products

kr Sym (Z0) × Mγr −k0 (a, ar+1)/R × ... k0 Sym (X)Mγ0 (a0, X, a1)× ks ... Sym (Z ) × Mγs−ks (as−1, b)

Where r ≤ 0 and 1 ≤ s. Again the moduli spaces M0(a, a) are ruled out. We must be careful because now the moduli space

M0(t, X, t) always contains the trivial connection and hence can be non-empty even when it has negative formal dimension. The formal dimension is −3b+(X) while the actual dimension is zero (again b1(X) = 0).

Tom Mrowka 18.937 The dimension of the above stratum

kr (Sym (Z0) × Mγr −kr (a, ar+1))/R × ...

k0 Sym (X) × Mγ0−k0 (a0, X, a1)× ks ... (Sym (Z ) × Mγs−ks (as−1, b))/R

∗ ∗ when a ∈ R (Y0) and b ∈ R (Y1)

s X + sfγ(a, b) + r − s + 1 − 4 ki − 3t + 3b (X). i=r Here s − r − 1 is the total number of breaks, t is the number of times the trivial connection appears in t the list ar+1,..., as−1 and  = 0 unless M0(t, X, t) appears in which case  = 1.

Tom Mrowka 18.937 Tom Mrowka 18.937 Now define ΦX : C∗(Y0) → C∗(Y1) by X X ˘ ΦX (a) = ]{[A] ∈ Mγ(a, X, b)}b. ∗ b∈R (Y1) sfγ (a,X,b)=0 Φ is a chain map. That we must check that

∂1ΦX + ΦX ∂0 = 0

Consider the end of Mγ(a, X, c) where now sfγ(a, X, b) = 1.

Tom Mrowka 18.937 The compactification in this case contains only those pieces where s X + 2 + r − s − 4 ki − 3t + 3b (X) ≥ 0 i=r So assuming b+(X) = 0 we must have r = −2, s = 1 or r = 0, s = 2 and in all cases ki = 0 and t = 0, i.e. ˘ Mγ−1 (a, a0) × Mγ0 (a0, X, b) or ˘ Mγ0 (a, X, a0) × Mγ1 (a0, b) The number of these ends is exactly the b component of ∂1ΦX (a) + ΦX ∂0(a) which is hence zero mod 2.

Tom Mrowka 18.937 p How does ΦX = ΦX depend on the choices (p) used in it constuction (Riemannian metric, perturbations...). These choices lie in a contractible space so we may join two choices by a path pt . Given P contained in the space of choice can form a moduli space P p Mγ (a, X, b) = ∪p∈P Mγ (a, X, b) If P is a smooth manifold of dimension d then assuming transversality P dim(Mγ (a, X, b)) = d + sfγ(a, X, b). We can now define a map

P ΦX : C∗(Y0) → C∗(Y1) by

P X X P ΦX (a) = ]{[A] ∈ Mγ (a, X, b)}b. ∗ b∈R (Y1) sfγ (a,X,b)=−d

Tom Mrowka 18.937 P Just as above if P has no boundary ΦX is a chain map. If P has boundary ∂P the formula gets modified to

∂P P P ΦX = ∂1ΦX + ΦX ∂0

P ∂P that is ΦX becomes a chain homotopy to zero for ΦX . Applying this to P = [0, 1] giving a homotopy pt between two different choices p0 and p1 we see that

po p1 P P ΦX + ΦX = ∂1ΦX + ΦX ∂0. and we see that the two maps are chain homotopic.

Tom Mrowka 18.937 The last general property we will need is a composition law.

Given X1 : Y0 → Y1 X2 : Y1 → Y2 then can construct maps ΦX1 and ΦX2 . We can also compose the cobordisms.

X12 = X1 ∪Y1 X2. and get a map

ΦX12 we then have the composition law

Φ + Φ ◦ Φ = ∂ ΦP + Φ0ΦP . X12 X2 X1 2 X12 X12

Tom Mrowka 18.937 P is a one parameter family as indicated in the figure.

Tom Mrowka 18.937 In particular as a corollary for any two choices p, p0 on a given 0 0 Y the chain complexes (Cp(Y ), ∂p) and (Cp (Y ), ∂p ) are chain homotopy equivalent.

Tom Mrowka 18.937 Construction of perturbations. Introduced by Taubes and used by Floer. Similar construction used by Donaldson. q : S1 × D2 → Y \ K smooth immersion. S1 = R/Z with s as coordinate. For each z in D2 let

holq(−,z)(B) ∈ (AdP)q(0,z) be the holonomy of the connection B around the corresponding ∗ loop based at q(0, z). Obtain a section holq(B) of q (AdP) over the disk D2.

Tom Mrowka 18.937 In terms of a trivialization the holonomy can solves the ODE

(˙g) = −ag

A solution to this equation is

∞ X Z g(t) = a(t0)a(t1) ... a(ti )dv∆i i=0 ∆i (t)

Here ∆k (t) is the scaled simplex

k X {(t0, t1,..., tk )| tj = t, tj ≥ 0}. j=0

Tom Mrowka 18.937 Pick

h : SU2 → R invariant under adjoint SU2 action. h induces also a function on q∗(AdP). Let µ be a non-negative 2-form supported in the interior of D2 and having integral 1, and define Z f (B) = h(holq(B))µ. (12) D2 A function f of this sort is invariant under the gauge group action.

Tom Mrowka 18.937 The formal gradient of such a cylinder function with respect to our L2 inner product on the tangent spaces of A. Let ∂h be the partial derivative of h after trivializing the cotangent bundle of SU2 using left-translation, we may regard this as a map

∗ ∂h : SU2 → su2.

Using the Killing form, we can also construct the su2-valued function (∂h)†. The SU2-invariance means that this also defines a map (∂h)† : AdP → adP. Let H be the section of q∗(adP) on D2 defined by

H = (∂h)†(holq(B)).

Tom Mrowka 18.937 We extend H to a section of q∗(adP) on all of S1 × D2 by using parallel transport along the curves s 7→ q(s, z): the resulting section H has no discontinuity at s = 0 even though the parallel transport around the closed loops may be non-trivial because of the SU2 invariance. The formal gradient of the cylinder function f , interpreted as a adP-valued 1-form on Y , is then given by

∗ (q)∗(Hµ). (13)

Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Two extensions. First allows non-trivial SO3-bundles second will incorporate knots. Choose and SO3 bundle P → Y . Let ω be a smooth 1-manifold 2 representative of w2(P) ∈ H (Y ; Z2). Consider connection 1 forms a ∈ Ω (Y \ ω) ⊗ su2 so that the holonomy of a asymptotically around tiny loops µ linking ω is −1 ∈ SU2.

Critical points are now

R(Y , ω) = {ρ ∈ Hom(π1(Y \ ω), SU2)|ρ(µ) = −1}/conj.

Tom Mrowka 18.937 Eg for T 3 and ω = pt × pt × S1, R(T 3, ω) is two points. We then looking for there matrices A, B, C so that [A, B] = −1, [A, C] = 1, [B, C] = 1. This implies C = ±1. [A, B] = −1 implies that upto conjugacy A = I and B = J (thinking SU2 as the unit quaternions). Let Σ be a Riemann surface of genus g. Lets consider representations ρ : π1(Σ \{p}) → SU2 with ρ(δ) = −1 where δ is a curve circling the puncture. Note that because −1 is in the center of SU2 this condition is base point independent. Choose generators of ρ : π1(Σ \{p}) so that g Y [αi , βi ] = δ i=1 2g Y R(Σ, p) = {(A1, B1,..., Ag, Bg) ∈ SU2 | [Ai , Bi ] = −1}/conj In particular such matrices never commute!.

Tom Mrowka 18.937 In using SO3 bundles we need generally to use the ”determinant 1” group of gauge transformations G1. The above choice of ω allows you keep track of things. We can now repeat the construction for SO3-bundles and we get

ω I∗ (Y ). and given a cobordism of pairs (X, ω) from (Y0, ω0) to (Y1, ω1) we get a map. ω ω0 ω1 ΦX : I∗ (Y0) → I∗ (Y1)

Tom Mrowka 18.937 As the T 3 example illustrates we can not have manifolds that ω have non trivial homology in particular I∗ (Y ) is well defined for all three manifolds so that there is an orientable surface S ⊂ Y meeting ω in an odd number of points.

Tom Mrowka 18.937 We can now get an interesting knot invariant originally introduced by Floer

Tom Mrowka 18.937 Given a link L ⊂ Y we can now choose and SO3-bundle on Y \ L and think of Y as a Z2-orbifold along L and representative 2 ω of w2(P) ∈ H (Y : Z2) ≡ H2(Y , ∂Y ; Z2). ω is now a

1-manifold with boundary on L. 1 Consider orbifold a ∈ Ω (Y \ ω) ⊗ su2 so that the holonomy of a asymptotically around tiny loops µ linking ω is −1 ∈ SU2 and around tiny loops ν linking L the holonomy is conjugate to

i 0  I = 0 −i

Tom Mrowka 18.937 Critical points are now

R(Y , L, ω) = 2 {ρ ∈ Hom(π1(Y \ ω), SU2)|ρ(µ) = −1 and ρ(ν) ∈ SI }/conj.

Note that if H ⊂ S3 is the Hopf link and ω is an arc connecting one component to another then we have

R(S3, H, ω) = pt.

We get the representation space we want if we take a link L and form the link L] = L ∪ H then.

3 ] ˆ R(S , L , ω) = RI(L).

The instanton homology construction still works and give a knot invariant I](L).

Tom Mrowka 18.937 We will want to deal with orbifolds, orbifold bundles and orbifold connections. We’ll only deal with Z2-orbifolds. X a Hausdorff space together with the following data. A collection of triples n (Γi , Vi , φi )Γi is either Z2 or the trivial group. Vi ⊂ R is a Γi invariant open subset and φ : Vi /Γi → X is a homeomorphism onto an open subset Ui . The open subsets Ui are an open cover. Further for each inclusion Ui ,→ Uj there is an injective immersion φij : Vi ,→ Vj which intertwines the Γi and Γj actions. (Note that there is a unique injective homomorphism Γi ,→ Γj and we’d have to be a little more careful here if the Γi had non-trivial automorphisms.) First for each (Γi , Vi , φi ) we have that Γi acts on Vi × G commuting with the G action. A orbifold principal bundle is the data of transition functions

τij : Vi × G → Vj × G which again intertwines the Γi and Γj actions. A connection in P is a connection in each Vi × G which is Γi invariant and natural under τij .

Tom Mrowka 18.937 We will consider only very special 4-dimensional Z2-orbifolds. We allow only one local model where the Z2 action is

(x1, x2, x3, x4) 7→ (x1, x2, −x3, −x4).

Let F be the fixed point (x1, x2, 0, 0). In this case the quotient 4 R /Z2 has a natural smooth structure so we can think of X as smooth four manifold with a distinguished two dimensional submanifold S. We’ll denote the orbifold by (X, S) though S might not globally be the fixed point set. S may or may not be orientable.

Tom Mrowka 18.937 Suppose P is an orbifold SU2-bundle and A is an orbifold connection in P. We get an bundle SU2 on X \ S with connectio. P may not extend over S. Suppose further that the Z2 action on P is non trivial. Then over F we get map F to Aut(SU2) = SO3. Any element of order 2 in SO3 conjugate to an inner automorphism by the element

1 0 0  I = 0 −1 0  . 0 0 −1

2 These inner automorphism form an RP ⊂ SO3.

Tom Mrowka 18.937 We can trivialize P so that the automorphism is locally constant Consider an invariant connection:  it z   it z  Γ + 1 1 dx1 + ... + 4 4 dx4. −z¯1 −it1 −z¯4 −it4 where t1, t2, z3, z4 are even under the involution while t3, t4, z1, z2 are odd. In particular along F the connection restricts to it 0  it 0  1 dx1 + 2 dx2 0 −it1 0 −it2

Tom Mrowka 18.937 4 Examples. 1. Let Z2 act on S by compactifying the standard action above. The fixed set is S2. The action lifts to an action 7 4 on the total space P± = S → S and leaves the standard 4 instanton (anti-instanton) invariant. X = S /Z2 is homemorphic 4 to S . The moduli space of Z2 invariant instantons is

3 H = SO3,1/SO3.

The action of such a connection 1 Z κ = ( ∧ ) = ± / 2 tr FA FA 1 2 8π X So for we have

d = 3, κ = 1/2, S · S = 0, χ(S) = 2, χ(X) + σ(X) = 2.

Tom Mrowka 18.937 2. Recall that CP2/conj = S4 (a folk observation made precise independently by Anrold, Kuiper, and Massey). Consider the 3 3 map f : C → Symo(R ). 1 f : 3 → z ⊂ S5 → <(zz¯t − |z|2). C 3

3 This is the real part of the moment map for SU3 acting on C . The image moment map consists of traceless hermitian matrices with exact 2 eigenvalues 2/3 and −1/3 for z ∈ S5. Then z is an eigenvector with eigenvalue 2/3 while if w is orthogonal to z the matrix acts by −1/3. Thus the image of the moment map is exactly CP2. Each of these authors show that 4 image of the real part is a copy of S . In fact let SO3 acts on traceless hermitian metrics by conjugation.

Tom Mrowka 18.937 This action evidently commutes with taking the real part. We can also diagonalize the real part by this action so we ask when given the diagonal entries of for which (a, b, c) does the (traceless) matrix  r ia ib   −ia s ic  −ib −ic 0

4 2 have a double eigenvalue. Call this orbifold (S , RP−).

Tom Mrowka 18.937 The Fubini-Study metric gives and ASD connection in ∗ 2 End0(T CP ).

∗ 2 2 2 2 p1(T CP ) = c1 − 4c2 = (9 − 12)H = −3H so k = 3/4. The dimension of this moduli space is

6 − 3(1 + 1) = 0 and eventually one can show this connection is unique upto gauge. The involution on CP2 is an isometry so lifts uniquely to ∗ T CP2. Pushing this down to the Note that

d = 0, κ = 3/8, S · S = −2, χ(S) = 1, χ(X) + σ(X) = 2.

Tom Mrowka 18.937 3. If we take the same example but reverse orientation we have 2 4 2 CP /Z2 = S . On CP there is a (unique upto gauge) reducible ASD connection in R ⊕ E where E is tautological bundle 2 − thought of on CP . Now p1(R ⊕ E) = −c2(C ⊕ E ⊕ E 1) = −1 so k = 1/4. Thus this moduli has formal dimension

8(1/4) − 3(1) = −1.

The −1 is accounted for by the stabilizer of the connection. There are two Z2 actions on R ⊕ E actions covering 2 conjugation on CP , these are ± conjugation on E and −1 on R. These however are gauge equivalent by multiplication by i in E. The resulting connection on S4 has

d = 0, κ = 1/8, S · S = 2, χ(S) = 1, χ(X) + σ(X) = 2.

Tom Mrowka 18.937 4. Next S2 × S2 with involution which is conjugation on each factor. The quotient is again S4. It is easy to see that the quotient is the union of two copies of D2 × D2 along their boundary. To see that we get the correct smooth structure think of S2 × S2 as the intersection of the cone

2 2 2 2 2 2 x1 + x2 + x3 − y1 − y2 − y3 = 0 with the unit sphere. The involution acts by x1 7→ −x1 and y1 7→ −y1 with the other coordinates fixed. Then we consider the map

f :(x1, x2, x3, y1, y2, y3) 7→ (x1y1, x2, x3, y2, y3).

This map descends to the quotient by the involution and on S2 × S2 the image of this map misses the origin so it can be rescaled to a map to S4.

Tom Mrowka 18.937 Furthermore if at least one of x1y1,, x2, x3, y2 or y3 is non zero then there are exactly two preimages unless we in the fixed set, since we must solve the equations

2 2 x1y1 = a, x1 − y1 = b with at least on of a or b non-zero. If both a and b are zero this means that x1 = y1 = 0, i.e. a fixed point of the involution.

Tom Mrowka 18.937 The involution acts by −1 on H2(S2 × S2), S = S1 × S1. Let L be line bundle which is the tensor product of the tautological bundle on the first factor and its dual on the second ∗ ∗ π1(O(−1)) × π2(O(1)).

Tom Mrowka 18.937 With the product metric and with both factors of the same area L admits and ASD connection indeed if ω1 and ω2 are the pullbacks under the two projections of the volume forms on the factors then −ω1 + ω2 is ASD under these assumptions. We use complex conjugation in either factor acting to lift of the involution to the total space of the line bundle. as our SO3-bundle take R ⊕ L, with the involution acting by −1 on the 2 R factor. p1(R ⊕ L) = c1 (L) = −2 so the dimension of the moduli space is 4 − 6 = −2

Tom Mrowka 18.937 Since the connection is reducible the deformation complex splits into the trivial part which has cohomology

0 2 1 H = H = R, and H = 0 account for all of the index. The L part of the complex has trivial cohomology. The involution acts by −1 on H0 and −1 × −1 = 1 on H2, thus corresponding orbifold connection on (S4, T 2) has h0 = 0, h1 = 0 and h2 = 0 thus we have

d = −1, κ = 1/4, S · S = 0, χ(S) = 0, χ(X) + σ(X) = 2

Tom Mrowka 18.937 4 2 4 2 4 2 4 2 (S , S ) (S , RP−) (S , RP+) (S , T ) κ 1/8 3/8 1/8 1/4 χ(S) 2 1 1 0 χ(X) + σ(X) 2 2 2 2 S · S 0 -2 2 0 d 0 0 0 -1 Table: Topological Invariants of basic examples

Tom Mrowka 18.937 Using excision one can see that the dimension the of the moduli space is an affine function of κ, χ(X) + σ(X), S · S, and χ(S) whether or not S is orientable:

3 d = 8κ − (χ(X) + σ(X)) + aS · S + bχ(S) + c 2 From the examples we see that a = 1/2, b = 1, c = 0 so we have Theorem

The formal dimension of the moduli space Mκ(X, S) is 3 1 d = 8κ − (χ(X) + σ(X)) + S · S + χ(S). 2 2

Tom Mrowka 18.937 Compactness theorem for orbifold connections. Since locally orbifold connections are just invariant connections there is nothing more to prove although when curvature concentrates along S, κ drops in units of 1/2 rather than 1. To describe the Uhlenbeck compactification on the pair (X, S) consider weighted collections of points

X 1 Sym (X, S) = { n x |n ∈ if x ∈ X\S and n ∈ if x ∈ S}. ∗ i i i Z i i 2Z i i

1 Then for κ ∈ 2 Z set X X Symκ(X, S) = { ni xi ∈ Sym∗(X, S)| ni = κ}. i i

Tom Mrowka 18.937 Thus

Sym 1 (X, S) = S 2

Sym1(X, S) = X ∪S Sym2(S)

Sym 3 (X, S) = (X × S) ∪S×S Sym3(S) 2 Sym2(X, S) = (Sym2(X) ∪ Sym2(S) × X ∪ Sym4(S))/ ∼ etc.

Tom Mrowka 18.937 Thus we have a compactification which now looks like Definition

The Uhlenbeck compactification UMκ(X, S) of Mκ(X, S) is the closure of [ Mκ(X, S) ⊂ Symκ0 (X, S) × Mκ−κ0 (X, S). 0 1 κ ∈ 2 Z with respect to the upto bubbling topology.

Tom Mrowka 18.937 An observation about representations of knot groups. Let

i 0  I = 0 −i

2 3 The conjugacy class of I is the equatorial SI ⊂ SU2 = S where ±1 are the north and south pole. Let

ˆ 3 2 RI(K ) = {ρ : π1(S \ K ) → SU2)|ρ(µ) ∈ SI }.

If U is the unknot then

ˆ 2 RI(U) = SI more generally ˆ 2 p RI(Up) = (SI ) taking homology recovers Khovanov groups (upto a grading shift)

Tom Mrowka 18.937 Indeed: ˆ 2 a 3 RI(Trefoil) = SI RP . The coincidence continues for 2-bridge knots (Lewallen). This is not a general fact but suggests are relationship between Khovanov’s groups and these representation spaces.

Tom Mrowka 18.937 Theorem Kronheimer-M 2009 For links L ⊂ S3 (with a distinguished base point) there is a spectral sequence starting with reduced Khovanov cohomology and abutting to I\(L).

Theorem \ Kronheimer-M If K is a knot I (K ) ≡ Z iff K ≡ U. The later theorem follows along the outlines work of Juhasz’´ sutured Floer Heegaard homology and transporting it to the instanton context. Corollary Reduced Khovanov homology detects the unknot.

Tom Mrowka 18.937 A couple of years ago Kronheimer and proved that there was a spectral sequence Kh(K †) =⇒ I](K ) (Study of I](K ) was motivated by Seidel and Smith and Wehrheim and Woodward, the spectral sequence by Ozsvath´ and Szabo´ and Bloom) I](K ) is only a mod 4 graded group while Khovanov homology is a Z ⊕ Z graded group. What remains of the homological and quantum gradings as they are ground through the spectral sequence machine? ] We’ll see that we can refine I (K ) to a Z ⊕ Z filtered group. (Slight in accurate, we’ll be more accurate later).

Tom Mrowka 18.937 It is functorial under cobordism (of pairs). If ` ∂(X, Σ, ω) = (−Y0, −K0, −ω0) (Y1, K1, ω1) then there is a map

Φ∗(X, Σ, ω): I∗(Y0, K0, ω0) → I∗(Y1, K1, ω1)

(well defined upto sign) which on the chain level is given by X Φ∗(X, Σ, ω)(α) = #(M(α, β))β β where now the sum is over zero dimensional moduli space on X.

Tom Mrowka 18.937 Exact sequences and spectral sequences Consider three knots, K2, K1, K0 which identical in the complement of a ball and in the ball looks like:

Such a ball is called a crossing. An orientation of K gives a crossing a sign. Tom Mrowka 18.937 These knots are cobordant via surfaces S2,1, S1,0, S0,2

Figure: The twisted rectangle T that gives rise to the cobordism S2,1 from K2 to K1.

These cobordisms induce maps

] Φ2,1 ] Φ1,0 ] Φ0,2 ... → I (K2) → I (K1) → I (K0) → ....

In fact this is an exact triangle.

Tom Mrowka 18.937 This follows from observing that for example

3 3 Z0 = (I × S , S2,1) ◦ (I × S , S1,0) is diffeomorphic to

3 4 2 Z1 = (I × S , S2,0)#(S , RP+). where S2,0 is the cobordism S0,2 viewed backwards.

T δ T2,1 δ 1,0

Figure: Curve δ has a neighborhood which is a mobius¨ band

Tom Mrowka 18.937 This means there is a one parameter family gt , t ∈ [0, 1] of metrics on Z. The two ends points of which realize the two decompositions above. For simple geometric reasons the moduli space for the metric g1 is empty. This means we can define a chain homotopy: X H0,2(α) = #(∪t Mgt (α, β))β β which verifies

Φ1,0 ◦ Φ2,1 = ∂0H2,0 + H2,0∂2.

These chain homotopies allow us to define a chain maps

C0 ↔ Cone(Φ2,1) given by (Φ0,2, H0,1) and (H2,0, Φ1,0). These maps are chain homotopy inverses proving the existence of the long exact sequence.

Tom Mrowka 18.937 4 2 To see that when we stretch out the (S , RP+) pair the moduli 4 2 space eventually becomes empty note that on (S , RP+) (this one is double branch covered by −CP2) we saw there was as unique irreducible solution. (S3, U) there is a unique flat connection and it is reducible with stabilizer U1. Thus when you stretch the moduli space becomes empty.

Tom Mrowka 18.937 The distguished triangle detection lemma. (Ozsvath and Szabo.)

Tom Mrowka 18.937 K 3 K0

K 2 K1

Figure: Two intersecting Mobius¨ bands, M31 and M20 inside S30.

Tom Mrowka 18.937 30

Y2 Y1

31 20

Figure: The five 3-manifolds, Y2, Y1, S30, S31 and S20 in the composite cobordism ([0, 3] × Y , S30).

Tom Mrowka 18.937 Q(Y2 ) Q(Y1 )

G'30

Q( 20) Q( 31)

Q( 30)

0 Figure: The family of metrics G30 containing the image of family G30.

Tom Mrowka 18.937 Tom Mrowka 18.937 Given a collection of crossings, X = {B1,... Bn} we can construct from K = K2...2, we obtain a collection of knots, Kv where v ∈ ZX. This family is 3-periodic in each component. We also have surfaces Svu which give cobordisms from Kv to Ku. When v = u + ei such surfaces induce maps just as before. More generally when v > u the surfaces come in families.

B1

B2

B3

τ3 τ1 τ2

I 3 The family of metrics Gvu parametrized by τ ∈ R (R in this example).

Tom Mrowka 18.937 Specializing to the case u, v ∈ {0, 1}X note that the family of ˘ metrics admits an action of translation by R. Let Guv denote the quotient by this action. We can define maps

Fvu : Cv → Cu by the formula

˘ X Gvu Fvu(α) = #(M )(α, β))β β where v > u, Fuu = ∂u and Fvu = 0 when v < u. We get a Khovanov like cube of resolutions.

(CX, FX) = (⊕uC,[Fvu]) This is a chain complex with differential F generalizing the mapping cone of the case (N=1). It is chain homotopy equivalent with (C(K ), ∂K ).

Tom Mrowka 18.937 If all of the resolutions Ku of X are unlinks we call X a \ pseudo-diagram. In this case I (Kv ) is is isomorphic to

2 2 H∗(×b0(Kv )S ) = ×b0(Kv )H∗(S ).

If in addition we have a diagram then the complex (CX, FX) is filtered by |v| and the E 1 page can be determined together with the differential. Then all the surfaces Svu with |v − u| = 1 are pairs of pants and the E 1 page of the spectral sequence is the Khovanov complex. A similar construction was done by Ozsvath and Szabo 2006. They related the Heegaard Floer homology of the double branched over of a link and gave a mod 2 relation between Khovanov homology and earlier and was a motivation for this work. Jonathan Bloom established a similar story for the Seiberg-Witten equations.

Tom Mrowka 18.937 ˘ The dimension of the moduli space MGvu (α, β) used in defining Fvu where α is the top critical point for Kv and β is the bottom critical point of Ku is 1 8κ + χ(S¯ ) + S¯ · S¯ + (dim G˘ ) vu 2 vu vu vu 1 = 8κ + χ(S¯ ) + S¯ · S¯ + χ(S ) − 1 vu 2 vu vu vu 1 ≡ χ(S ) + S¯ · S¯ − 1 (mod 2). vu 2 vu vu

Here κ = 1 R tr(F ∧ F ) and the last equality use the 8π2 R×S3 A A second bullet point. κ ≥ 0 if the moduli space is non-empty. 1 ¯ ¯ 1 Furthermore κ = 16 S · S (mod 2 ).

Tom Mrowka 18.937 3 ¯ 4 Given a cobordism Svu ⊂ I × S with v > u define Svu ⊂ S to be the closed surface obtained by capping off each unknot ¯ ¯ component by a disjoint disk. Define σ(v, u) = Svu · Svu. Extend σ(v, u) to (Zn)2 so that σ(v, u) = σ(v, w) + σ(w, u).

Note that given an orientation of K there is a preferred smoothing, the oriented one and so a distinguished vertex of the cube o ∈ {0, 1}X.

In order for the homological gradings for two pseudo-diagrams to have anything to do with each other we need to introduce a grading shift. For a crossing set X define the h-grading of the summand Cv to be X 1 h = − v(c) + σ(v, o) + n 2 − c∈{0,1}X

Tom Mrowka 18.937 With this choice FX increases the filtration level by at least one for every element of CX. With this definition of h for all v > u we have that 1 dim(M ) ≡ χ(S ) + S¯ · S¯ − 1 (mod 2) Gvu vu 2 vu vu ! X 1 ≡ (v(c) − u(c)) − σ(v, u) − 1 (mod 2) 2 c∈N

≡ h|Cu − h|Cv − 1 (mod 2) so Fvu has odd degree (computed from dim=0 moduli spaces). We also we have for all v > u that

ordhFvu ≥ h|Cu − h|Cv ! X 1 = (v(c) − u(c)) − σ(v, u) ≥ 0 2 c∈N ordhFvu ≥ 1.

Tom Mrowka 18.937 Dropping and adding crossings Suppose we have two 0 ∗ psuedo-diagrams X and X = X ∪ {c }. Then we have complexes CX and CX0 . We can write CX0 = C0 ⊕ C1 as a group. Also introduce C−1 and C2 obtained from the knots Kv , c(v) ∈ {0, 1} for c 6= c∗ and c(v) = −1 or 2. As groups these 0 0 are CX but with the grading h given by X .

Analogous to the discussion of the differential there are chain homotopy equivalences Φ: C2 → CX0 and Ψ: CX0 → C−1 which each increase filtration by at least one and the requisite chain homotopies are non-deceasing on the filtration level.

Tom Mrowka 18.937 0 Now we can compare the h and h filtrations on CX,C−1 and C2. For C2, we have 1 h0 − h = v(c∗) + (σ(v 0, o0) − σ(v, o)) + −n + n0 2 − − 1 1 = 2 + (σ(v 0, o0) − σ(v, o)) + (1 − ). 2 2 where  = 1 if c∗ is a negative crossing and 0 if c∗ is positive. For a positive crossing we have σ(v 0, o0) − σ(v, o) = 2 and 1 −  = 0, while for a negative crossing we have σ(v 0, o0) − σ(v, o) = 0 and 1 −  = 2. In either case the difference is 1.

Thus as a map from the h0 filtration to the h filtration Φ has non-negative order. Similarly Φ can be seen to have non-negative order.

Tom Mrowka 18.937 This discussion leads to a simple proof that the h-filtered complexes are invariants (upto filtered chain-homotopy equivalence). For example we can view Reidermiester III as a composite of dropping crossing, isotopy and adding crossing.

1 The map induced by cobordisms in general has order ≥ 2 S · S.

Tom Mrowka 18.937 This discussion extends to the quantum filtration though it is more subtle. For example the q-filtration on Cv is ! X 3 q = Q − v(c) + σ(v, o) − n + 2n 2 + − c∈X

2 where Q is the degree of a generator in H∗(×b0(Kv )S ) minus b0(Kv ).

Tom Mrowka 18.937 1 Q(β ) − Q(β ) = 8κ + χ(S) + (S · S) + dim(G ) − 1 0 1 2 vu jS · S k ≥ χ(S) + S · S − 4 + dim(G ) − 1 8 vu ≥ χ(S) + S · S + dim(Gvu) − 1 (If S.S < 8) hence 1 X q(β ) − q(β ) ≥ − (S · S ) + (v(c) − u(c)) − 1 ≥ −1 0 1 2 vu vu c But parity of q depend only on number of components of K .

Tom Mrowka 18.937 Empirically the spectral sequence tends not to have many other differentials. The smallest knot for which it does not collapse is the (4, 5)-torus knot. I](K ) sees knot genus and whether the knot fibers (as do Heegaard Floer and Seiberg-Witten knot groups). Does Khovanov homology do so?

Tom Mrowka 18.937 Tom Mrowka 18.937 The unknotting problem.

Which one is unknotted?

Tom Mrowka 18.937 Goertiz’ hard unknot (the second one).

Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Some basic questions in three/four manifold topology: Recognize different three manifolds and links in them.

If H2(Y ) or H2(Y ; ∂Y ) are non-zero find the most efficient representative. E.g. every knot K in S3 bounds an embedded oriented surface. What is the minimal genus of of such surfaces. This is called the Seifert genus g3(K ). Decide if a 3-manifold fibers over S1. Decide if a 3-manifold admits a geometric structure. Decide a 3-manifold bounds a four-manifold with certain properties for example does a homology sphere bound a homology ball. If K ⊂ Y and Y = ∂X and K is null homologous in X what is the minimal genus of such a surface. E.g. K ⊂ S3 = ∂B4. The minimal genus is called the four-ball genus g4(K ).

Tom Mrowka 18.937 An important tool coming from studying various PDEs arising in high energy physics is Floer homology. This give invariants of 3-manifolds and knots in them.

(Y , K ) F∗(Y , K ). The grading ∗ might belong to an affine space (like Spinc-structures) for a finitely generated abelian group.

These invariants are functorial under cobordism a ∂(X, S) = (Y−, K−) (Y+, K+) then F∗(X, S): F∗(Y−, K−) −→ F∗(Y−, K−)

Tom Mrowka 18.937 In historical order Instanton Floer homology (Floer, 1987) I∗(Y ), ∗ ∈ Z8 Seiberg-Witten Monopole Floer homology (1994, many authors, KM for complete treatment) HM∗(Y ), ∗ ∈ π0(Non-vanishing vector fields) Heegaard Floer homology (Ozsvath-Szabo, 2002) HH∗(Y ) ∗ same as HM Heegaard Knot Floer homology. (Ozsvath-Szabo, Rasmussen, 2004) Embedded Contact homology (Hutchings 2002, Hutching-Taubes 2008) ECH(Y , ξ), ∗ same as HM.

Un-Instanton Floer homology, Kronheimer-Mrowka (2007) I∗(Y , n), ∗ ∈ Z4n Quilted Instanton Floer homology. Woodward and Wehrheim. (2008) QI∗(Y ). SW , HH and ECH are now know to be isomorphic (or at least proofs have been announced) while the Instantons version are different. Tom Mrowka 18.937 By taking graded Euler characteristics the knot invariants coming from these theories are related to the Alexander polynomial and its generalizations, the Milnor-Turaev torsions for knots in general 3-manifolds. Eg for a knot of one 3 3 component K ⊂ S and for HFK∗(S , K ) ∗ ∈ Z × Z and

X i j 3 ∆K (t) = (−1) t rank(HFKi,j (S , K )). i,j

Tom Mrowka 18.937 It turns out that many of various Floer homology theories are quite effective at the several problems. In fact using Floer homology one can Detect the unknot. Determine the minimal genus of embedded surfaces. Distinguish many three manifolds. Provide constraints on the topology for 4-manifolds bounded by a given 3-manifold. Detect if 3-manifolds are fibered. The relationship between these invariants and the geometrization of three manifolds remain unclear.

Tom Mrowka 18.937 There are some very different kinds of knot and 3-manifold invariants. Those coming from Quantum groups and the like the Jones Polynomial being the premier example (then there are the Reshetikin-Turaev and Turaev-Viro invariants etc). In 1998, Mikhail Khovanov found a knot invariant

Khi,j (K ) like the ones above that takes the form of a bi-graded vector space so that again taking the Euler characteristic in one grading and Poincare´ polynomial in the other gives the Jones-Polynomial.

1 X i j i,j J(t)| 1/2 = (−1) q rank(Kh (K ). t =−q q + q−1 i,j

Tom Mrowka 18.937 Generally it has been more difficult to find applications of these invariants to low dimensional topology though there are some including estimates of the Thruston-Bennequin number of knots in the standard contact S3. The most striking application is however Ramussen’s s-invariant. This invariant can be used to estimate the four-ball genus of a knot and gave the first purely combinatorial proof of Milnor’s conjecture on the unknotting number of torus knots first proved in 1993 by Kronheimer-M.

Tom Mrowka 18.937 Review of construction of Khovanov cohomology.

The basic algebraic object is an 2-dimensional vector space with V with two generators v+, v−. V also has two operations. A multipication m : V ⊗ V → V with v+ as the unit and m(v− ⊗ v−) = 0. There is also a comultiplication ∆ : V → V ⊗ V with ∆(v+) = v− ⊗ v+ + v+ × v− and ∆(v−) = v− ⊗ v−. V has a quantum grading where gr(v+) = 1 and gr(v−) = −1.

Tom Mrowka 18.937 (A, m, ∆) can be used to define a 1 + 1 dimensional Topological Quantum Field Theory. a Kh( S1) = V ⊗p. p For the elementary cobordism

P = set Kh(P) = m. For the elementary cobordism

Q = set Kh(Q) = ∆. ` 1 ` 1 More generally given a cobordism from p S to q S define a map by cutting it up into elementary cobordism and composing the resulting maps. The TQFT name applies here because this composition map is independent of how you cut things up. Tom Mrowka 18.937 Next given a knot diagram of N crossings:

Form the 2N cube of resolutions according to the rule:

Tom Mrowka 18.937 For example for the Hopf link the cube is:

Tom Mrowka 18.937 For example for the Hopf link the cube is:

Tom Mrowka 18.937 Using the TQFT to define a differential we can compute the homology of the resulting complex which the Khovanov homology. Independent of the diagram.

In fact if we orient the link there is a bigrading. Set n+ to be the number of positive crossings and n− to be the number of N negative crossings. For v ∈ {0, 1} put Kh(Kv ) in homological grading |v| − n+ and shift the quantum grading by |v| + 2n−n+.

0 0 0 0 0 0 0 Z 0 0 0 0 0 Z 0 0 0 0 Z Kh(U) = 0 0 0, Kh(Trefoil) = 0 0 0 0 0 Z 0 0 Z 0 0 0 0 0 0 Z2 0 0 0 Z 0 0 0

Tom Mrowka 18.937 This lead to the fact that FI∗(Y , K ) is an invariant of unoriented links. This theory has some nice generalizations. The fundamental group of the configuration space means we can introduce twisted coefficient groups. This is analogous to the Lee perturbation of Khovanov homology. It can be used to re-recover many results on slice genus for example for algebraic knots.

It generalizes to SUn gauge theory and appears to be related of Khovanov-Rozansky theory there.

Tom Mrowka 18.937 Floer’s Instanton Knot Floer Homology

Upto isomorphism I∗(Y , ω) depends only on the homology class of ω. Can make a interesting invariant of a (framed) knots in Y , K˜ = S1 × D2 ⊂ Y as follows. Form

Z(Y , K˜ ) = Y \ S1 × D2 ∪ (T 2 \ D2) × S1

with ω = point × γ. Set ˜ ˜ IK(Y , K ) = I(Z(Y , K ), ω).

If K ⊂ S3 is a knot we’ll write

3 ˜ IK(K ) = IK(S , K )

where K˜ here is the knot with its canonical framing.

Tom Mrowka 18.937 Properties of Instanton Knot Floer Homology. rk(IK(K )) = 1 iff K is the unknot. µ(Σ) sTr(t : IK(K ) → IK(K )) = ∆(t) (Cf Meng-Taubes for SW, Ozsvath-Szabo for Heegaard Floer) The dimension of top eigenspace is 1 iff K is fibered. (Cf Ghiggini, Ni for Heegaard, Floer.) NOT functorial under cobordism of pairs! These properties are proved using a further extension to sutured manifolds. These are three manifolds with boundary decorated by a collection of closed curves on their boundary. (cf Juhasz for Heegaard Floer).

Tom Mrowka 18.937 Singular Instanton Floer Homology. We can generalize this discussion following work that Kronheimer and I did in the early 90’s. We consider connections on Y \ K which look like

−1 0 A + iα η 0 1

A is a smooth connection on Y and η is a smooth one from on Y \ K restricting the angular 1-form dθ near K . The holonomy on loop linking K is exp(2πiα) 0  0 exp(−2πiα).

The critical points of the Chern-Simons function are the conjugacy classes of representations

ρ : π1(Y \ K ) → SU2 with holonomy on loops linking K as above.

Tom Mrowka 18.937 The choice of conjugacy class above is significant in being able to avoid the Novikov ring situation. The configuration space relevant to this story has fundamental group Z2 and for a general conjugacy class the periods of CS are

2 ∆CS = 8π (k + 2αl) and for the spectral flow is

∆sf = 8k + 4l.

Thus when α = 1/4 these are proportional. We are still not ready to construct a Floer theory. It turns out that because of the non-free action of G the singularities in A/G ruin the Morse theory construction.

Tom Mrowka 18.937 The Khovanov complex of a planar diagram D (C = C(D), dKh) is a bi-graded complex

M i C(D) = Cj(D) i,j obtained from the cube of resolutions. The differential satifies

i, i+1 dKhCj (D) ⊂ Cj The Khovanov cohomology, the cohomology of this complex is then a bi-graded group.

i,j i i+1 i−1 Kh (K ) = ker(dKh : Cj → Cj )/(dKh(Cj )

Tom Mrowka 18.937 Cob3,1 whose objects consist of triples (Y , L, ω) where Y is a connected oriented non empty three manifold. L ⊂ Y is a unoriented, possibly empty link in Y ω ⊂ Y is a one manifold with boundary. The boundary points are only on L and there is surface Σ ⊂ Y \ L which meets ω tranversally in an odd number of point. To each such pair we associate a finite dimensional Z-module

I∗(Y , L, ω) the instanton homology. These groups are functorial under the natural notion of cobordism.

Tom Mrowka 18.937 Getting knot invariants.

A knot K ⊂ S3 can naturally be promoted to an object in our category

(S3, K )(S3, K #, ω) Define I](K ) = I(S3, K #, ω)

Tom Mrowka 18.937 Theorem Kronheimer-M 2010 For links L ⊂ S3 (with a distinguished base point) there is a spectral sequence starting with Khovanov homology and abutting to I#(L).

Theorem # Kronheimer-M If K is a knot I (K ) ≡ Z iff K ≡ U. The later theorem comes from slightly earlier work carrying over Juhasz’´ sutured Floer homology to the instanton context. Corollary Reduced Khovanov homology detects the unknot.

Tom Mrowka 18.937 Notice that since in SU2 an element is conjugate to its inverse there choice of co-orient inherent in the definition of the configuration space via the choice of η is actually irrelevant. This lead to the fact that FI∗(Y , K ) is an invariant of unoriented links. This theory has some nice generalizations. The fundamental group of the configuration space means we can introduce twisted coefficient groups. This is analogous to the Lee perturbation of Khovanov homology. It can be used to re-recover many results on slice genus for example for algebraic knots.

It generalizes to SUn gauge theory and appears to be related of Khovanov-Rozansky theory there.

Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 How can we detect or certify unknots?

The simplest way is to specify a embedded disk which the knot bounds. In the 1961 Haken used this characterization to give the first algorithm to detect the knottedness. Input a polygonal knot. Triangulate the complement. Look for piecewise linear surfaces that bound the knot amongst an exponentially large collection. This uses Haken’s theory of normal surfaces. This is Check if any of these are disks.

Tom Mrowka 18.937 1998 Hass, Lagarias and Pippenger analyze Haken’s algorithm carefully and show. It runs in exponential time in either number of crossings or number of tetrahedra used in triangulating the complement. (Uses Jaco and Oertel’s simplification of Haken’s algorithm) Detecting the unknot is an NP problem. This uses a further simplification due to Jaco and Toeffelson).

Tom Mrowka 18.937 Algebraic characterization of unknottedness.

3 K ≡ U iff π1(S \ K ) ≡ Z.

Thurston’s geometrization for Haken manifolds (including knot 3 complements) implies π1(S \ K ) is residually finite. That is that every nontrivial element is detected by a map to a finite group. This leads to another algorithm to detect the unknot. Find

Tom Mrowka 18.937 Birman-Hirsch (1998) and Dynnikov (2000) algorithms. Roots in Bennequin’s analysis of singular foliations of surfaces induced by the standard contact structure in R3. Dynnikov’s algorithm works with knots represented as grid diagrams.

Tom Mrowka 18.937 All horizontal line go under the vertical.

Tom Mrowka 18.937 There are three moves you can do on a grid diagram.

Cyclic permutation of horizontal edges. Exchange of horizontal edges if the don’t interleave. Stablization or Destablization.

Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Tom Mrowka 18.937 Dynnikov shows that if you are given a gird diagram of the unknot you can reach the square diagram via cyclic horizontal permutations, exchange moves and destablizations (no stablizations). The number of vertical lines does not increase in this process. This leads to another algorithm which again involves exhaustive search and exponential time.

Tom Mrowka 18.937 The Jones polynomial and knot homology theories.

The introduction of more powerful knot invariants gave rise to the hope that other invariants might be able to recognize unknots. The Jones polynomial introduced in 1984 is characterized by

VU (t) = 1 and 1 1 √ VK − tVK (t) = (√ − t)VK (t) t + − t 0

K+ K- K0

Tom Mrowka 18.937 Because of its recursive nature the Jones polynomial is straightforward to compute. It is however not computationally cheap. The problem of computing the value of the Jones polynomial at certain points is know to be P#-hard (Jaeger, Vertigan and D. J. A. Welsh). It is unknown if the Jones polynomial detects the unknot.

Tom Mrowka 18.937 In the late 1980’s Floer introduced his Floer homology groups. Application to symplectic topology and invariants of three manifolds. The invariants are the Morse homology of the Chern-Simons function three manifolds. The construction exploited Instantons. TQFT-like flavor. To each three manifold Y he associates a finite dimensional vector space

I(Y ).

To each cobordism W : Y0 → Y1 he associates a map

I(W ): I(Y0) → I(Y1).

For homology spheres I(Y ) geometrified Casson’s invariant

χ(I(Y )) = 2λ(Y ).

Tom Mrowka 18.937 In 1994 the Seiberg-Witten invariant where introduced. Kronheimer-M used the associated Monopole Floer homology invariants, HM˜ (Y ). to prove Theorem 3 3 Let S0(K ) denote 0-surgery on a knot K ⊂ S . Then ˜ 3 HM(S0(K )) = 0 if and only if K is the unknot.

Indeed we showed that these invariants detect the minimal genus of representatives of homology classes in Y 3.

Tom Mrowka 18.937 Two problems. Computing HM˜ was not known to combinatorial so we don’t get an algorithm. The proof, while geometrically elegant, used symplectic and contact geometry coming from Eliashberg-Thurston and used Gabai’s existence theorem for foliations. Gabai’s existence theorem used his sutured manifold hierarchies. Would be nice to get this more directly.

Tom Mrowka 18.937 Khovanov cohomology. In 1998 Khovanov introduced his homology groups for knots. To each planar oriented diagram of a knot, D he associated a chain complex (C(D)∗,∗, d)

d(C(D)i,j )C(D)i+1,j .

Cohomology groups are invariants of the knot:

Kh∗,∗(K ).

The Khovanov homology categorifies the Jones polynomial. X ( )|√ = ( + −1) (− )i j ( i,j ( )). VK t (t)→−q q q 1 q rank Kh K i,j

Tom Mrowka 18.937 The construction is purely combinatorial and is again easy to program. Bar Natan and Shumakovich have rather fast program that can handle pretty large knots. It is functorial for cobordisms of knots. Rasmussen exploited this to give the first combinatorial proof of the Milnor conjecture on unknotting numbers of torus knots, one of the first applications of gauge theory to knot theory (Kronheimer-M 1993). It is not known to generalize to an invariant of knots in three manifolds. Cube of resolutions.

Tom Mrowka 18.937 Heegaard Floer homology Beginning around 2000 Ozsvath´ and Szabo´ began developing their version of Floer homology. Original version was based holomorphic geometry of a Heegaard splitting. Much more computable than Seiberg-Witten or Instanton Floer homology groups. Ozsvath´ and Szabo´ and independently Rasmussen introduced HFK an invariant of knots in S3. They show it detect knottedness and in fact knot genus (2006) Ni showed that HFK detects fibering (Following work of Ghiggini) (2007) Juhasz´ showed how to generalize Heegaard Floer theory to sutured three manifolds and how to exploit the suture manifold hierarchy directly.

Tom Mrowka 18.937 Double branched covers. Let D(K ) denote the double branched of S3 branched along the knot K . Ozsvath´ and Szabo´ proved the following. Theorem

2005 There is a spectral sequence whose E2-page is the reduced Khovanov homology

∗,∗ Kh˜ (K )

∗ and converges to HH (D(K )). (Everything with Z2 coefficients). Sadly if K is the (3, 5)-torus knot then D(K ) is the Poincare´ homology sphere and HH∗(D(K )) = HH∗(D(U)) so it is not an unknot detector.

Tom Mrowka 18.937 HFK is combinatorial.

In 2008, Ozsvath,´ Manolescu and Sarkar showed that HFK (K ) could be computed completely combinatorially from a grid diagram. In particular they give another combinatorial algorithm that detect knottedness (genus, fibering). Their complex is however quite huge n! generators with many differentials. It is programmable.

Tom Mrowka 18.937 Cob3,1 whose objects consist of triples (Y , L, ω) where Y is a connected oriented non empty three manifold. L ⊂ Y is a unoriented, possibly empty link in Y ω ⊂ Y is a one manifold with boundary. The boundary points are only on L and there is surface Σ ⊂ Y \ L which meets ω tranversally in an odd number of point. To each such pair we associate a finite dimensional Z-module

I∗(Y , L, ω) the instanton homology. These groups are functorial under the natural notion of cobordism.

Tom Mrowka 18.937 Getting knot invariants.

A knot K ⊂ S3 can naturally be promoted to an object in our category

(S3, K )(S3, K #, ω) Define I](K ) = I(S3, K #, ω)

Tom Mrowka 18.937 Theorem Kronheimer-M 2010 For links L ⊂ S3 (with a distinguished base point) there is a spectral sequence starting with Khovanov homology and abutting to I#(L).

Theorem # Kronheimer-M If K is a knot I (K ) ≡ Z iff K ≡ U. The later theorem comes from slightly earlier work carrying over Juhasz’´ sutured Floer homology to the instanton context. Corollary Reduced Khovanov homology detects the unknot.

Tom Mrowka 18.937 Let me indicate how a Pin−-structure is useful for non-orientable surfaces. First of all the data we need to specify the story in the orientable case is a line bundle L → S and rank two vector bundle E → W and an isomorphism

−1 L ⊕ L → E|S.

Using this isomorphism a connection in L induces a reducible connection A in E on a neighborhood of S. Then we can add

−1 0 A + iα η 0 1 as before.

Tom Mrowka 18.937 A Pin−-structure on S gives us a rank two vector E → S which when a local orientation is specified is decomposed as E + ⊕ E − and the Levi-Civita connection on S induces a connection in A in E respecting this local splitting. We can still write

−1 0 A + iα η 0 1 for our local model. This means now that when we pick a local orientation of S we get both a splitting of E and a local normal orientation defining η. If we switch local orientation both the splitting switches and η changes sign so the above expression is invariant.

Tom Mrowka 18.937 The Seifert genus determines if a knot is trivial or not. The slice genus does not. There is a difference between smoothly slice and topologically (locally flat) slice! I will focus mainly on the smooth sliceness question. Fox’s Slice vs. Ribbon question. Minimal genus problem in dimension four.

Tom Mrowka 18.937 2g Let S be an oriented Seifert surface. H1(S) = Z . Choose a curves a1,..., a2g which give a basis. Define the Seifert Matrix to be + V = [lk(ai , aj )]. t ∆k (t) = det(V − tV ) Is the alexander polynomial. A classical invariant is the knot signature σ(K ); the signature of the matrix V + V t . The signature bounds the slice genus

2g4(K ) ≥ |σ(K )|.

Not a bad invariant but σ(T (5, 4)) = 8 and g4(K ) = 6

Tom Mrowka 18.937 The Casson-Gordon invariant (circa 1974) this is an invariant stemming from the G-signature theorem. The big advance at the time was the non-sliceness of most twist knots.

Tom Mrowka 18.937 The Anti-Self-Dual equations through the Donaldson invariants figured prominently in the next phase of development. The slice genus of many knots was determined. Related invariants due to Fintushel and Stern settled the case of many Pretzel knots, Matic,´ and Ruberman generalized these ideas to give finer invariants.

Tom Mrowka 18.937 A sample theorem from that time is Theorem (Kronheimer-Mrowka) Let C ⊂ B4 ⊂ C2 be a ”piece of a plane curve”. Suppose that C intersects ∂B4 transversally in a knot K . Then g4(K ) = genus(C). For example this implies that the slice genus of the (p, q)-torus knots is 1 (p − 1)(q − 1). 2 The introduction of the Seiberg-Witten equations in 1994 lead to simpler proofs of these results. Proof of Thom conjecture.

Tom Mrowka 18.937 In 2001-2002 in a wonderful series of paper Ozsvath´ and Szabo´ introduced a “Floer homology” type invariant which associates to any knot in S3 a vector space. From this in 2003 they found a concordance invariant τ(K ) which satisfies |τ(K )| ≤ g4(K ). τ(K ) is rather computable. Earlier this year Manolescu, Ozsvath,´ Sarkar showed that the version for knots and links in S3 can be obtained by purely combinatorial means. Heegaard Floer homology is part of package of invariants of three and four-dimensional smooth manifolds.

Tom Mrowka 18.937 In 1998 Khovanov introduced another invariant which associates to any knot in S3 a vector space. From this in 2004 Jake Rasmussen found another concordance invariant s(K ) which satisfies

|s(K )| ≤ 2g4(K )

σ(K ) is also rather computable and has a completely combinatorial defintion. Hedden and Ordung showed that there are knots for which |s(K )| < 2|τ(K )| and 2|τ(K )| < s(K )|

Tom Mrowka 18.937 Tom Mrowka 18.937 Singular Instanton Floer homology is functor from the category Cob3,1 whose objects consist of triples (Y , L, ω) where Y is a connected oriented non empty three manifold. L ⊂ Y is a unoriented, possibly empty link in Y ω ⊂ Y is a one manifold with boundary. The boundary points are only on L and there is surface Σ ⊂ Y \ L which meets ω tranversally in an odd number of point. Morphisms are again triples (X, Σ, ω) consisting of a four-manifold, X with boundary, embedded surface Σ transverse to the boundary, and ω a embedded surface with corners with boundary contained in ∂X ∪ Σ. To each object of we associate a finite dimensional Z-module

I∗(Y , L, ω).

Morphisms induce maps

Φ(X,Σ,ω) I∗(Y0, L0, ω0) −→ I∗(Y1, L1, ω1).

Tom Mrowka 18.937 Let (C, ∂) be a differential group. Z2 (descreasing) filtration is i,j 0 collection F C ⊂ C so that Fi,j C ⊂ Fi0,j0 C when ever i ≤ i and j ≤ j0.

M C = Ci,j. i,j

Let COMPZ2 be the category with

2 Obj(COMPZ2 ) = Z − filtered differential groups.

and Mor(COMPZ2 ) consist of chain maps

φ : A → B Tom Mrowka 18.937 Review of ”singular connections”.

X a closed four-manifold and S ⊂ X a closed surface. Consider connections on X \ S which look (locally) like

−1 0 A + iα η 0 1

A is a smooth connection on X and η is a smooth one from on X \ S restricting to a smooth angular 1-form dθ on the punctured fibers Nx \ x of the normal bundle N → S The holonomy on a shrinking family of loops linking S is asymptotic to exp(2πiα) 0  . 0 exp(−2πiα)

Tom Mrowka 18.937 We restrict to α = 1/4, i.e. holonomy conjugate to

i 0  I = . 0 −i

2 SI ⊂ SU2 is the sphere of conjugates of I. Upto conjugacy this doesn’t depend on the orientation of the normal loop, S can be non-orientable.

Then as an SO3-connection we can think of this as an orbifold connection on Xˆ , which is X viewed as an orbifold with a cone angle π along S.

N.B. P → Xˆ does not necessarily give rise to a bundle on X. however there is a double cover ∆ → S and P extends to ∆ and reduces to an O2-bundle there. We keep track of w2 via ω ⊂ X \ S a perhaps non-orientable surface with ∂ω ⊂ S.

Tom Mrowka 18.937 We’ll look at moduli spaces Mω of orbifold connections A on orbifold bundle P → X with w2 represented by ω solving the ASD equations: + FA = 0. The action and dimension formula. As usual we set 1 Z κ = ( ∧ ∗ ) ≥ 2 tr FA FA 0 8π X for A ASD. 1 κ = S · S (mod 1)/2. 16 Then [ Mω = Mκ,ω κ The expected dimension of Mκ,ω is 3 1 8κ + (χ(X) + σ(X)) + χ(S) + S · S. 2 2

Tom Mrowka 18.937 Instanton Floer Homology

With these equation in mind we consider the case of X = R × Y and S = R × K for some link K ⊂ Y . Consider ˆ P → Y an orbifold principal SO3 bundle. Keep track of w2(P) by a choosing a one manifold ω ⊂ Y with ∂ω ⊂ K .

ASD equation on R × Y downward gradient flow for Chern-Simons function.

Critical points of CS mod gauge ρ : π1(Y \ (K ∪ ω)) → SU2 s.t. ρ(∂∆) = −1 where ∆ ∩ ω = pt 2 and ρ(∂∆) ∈ SI when ∆ ∩ K = pt. mod conjugation. We want to avoid reducible representations. Enough if there is Σ ⊂ Y so that all reps of π1(S \ S ∩ (K ∪ ω)) with given conditions are irreducible. If k = ](S ∩ K ) and m = ](S ∩ ω) we need ±k + 2m 6∈ 4Z. Eg.k = 0 and m = 1 is OK. Tom Mrowka 18.937 The Morse complex is M C∗(Y , K , ω) = Λ(α) α∈Crit(CS) with boundary map X ∂(α) = #(M(α, β))β β where the sum over moduli space of solutions which are 1-dimensional. The homology of CS on Aα/Gα is denoted

I∗(Y , K , ω).

This a Z4 graded Z-module.

Tom Mrowka 18.937 More generally the graded tensor algebra of H∗(X), A(X) acts. Given zX ∈ A(X) there is a map

µ(zX ): I∗(Y0, ω0) → I∗(Y1, ω1)

In particular H2(Y ) = H2([−1, 1] × Y ) acts on I∗(Y , ω). Given Σ ⊂ Y a closed oriented embedded two manifold then

Spec(µ([Σ])) ⊂ (χ(Σ), −χ(Σ)) ∩ 2Z.

Recovers the Thruston norm on H2(Y ). Σ minimizes genus in its homology class if and only if

χ(Σ) ∈ Spec(µ([Σ])).

Tom Mrowka 18.937 A few years ago Peter Kronheimer noticed that if we look at the 3 space of representations of of π1(S \ K ) into SU(2) where the meridan of knot goes to the conjugacy class of

i 0  I = 0 −i in these examples we get 2 2 S for the unknot or ×N S for unknot of N-components. 2 ` 3 S RP of the trefoil. Notice that the homology of these spaces reproduces the Khovanov homology of this examples.

Tom Mrowka 18.937 Getting knot invariants. A knot K ⊂ S3 can naturally be promoted to an object in our category

(S3, K )(S3, K \, ω) and

(S3, K )(S3, K ], ω) Define I\(K ) = I(S3, K \, ω) and I](K ) = I(S3, K ], ω)

Tom Mrowka 18.937 We’ll also consider the borderline case A1 and G2. G2 is not a 2 Banach Lie Group. Define the L1-length of a path γ :[0, 1] → ALn to be Z 1 dγ `(γ) = k kL2 dt 0 dt 1,γ(t) 2 ∞ then define the L1-distance between two C connections to be

d(A0, A1) = inf `(γ). γ:γ(i)=Ai

Lemma

For all A0 there is an  > 0 and ρ > 1 so that for all A ∈ A1 with kA − A0k 2 ≤  we have: L1 1 d(A, A0) ≤ kA − A0kL2 ≤ ρ d(A, A0). ρ 1

2 The quotient space B1 inherits a notion of L1-distance by setting ˆ Tom Mrowka 18.937 d([A0], [A1]) = inf {d(g · A0, A1)}. g∈Gs+1 Here is a standard lemma from the gauge theory literature (see [?] Appendix A.) Lemma dˆ is a metric. The induced topology is the quotient topology. The fact that dˆ satisfies the triangle inequality is general nonsense. To prove that dˆ is a metric must show that if ˆ d([A0], [A1]) = 0 then [A0] = [A1]. If the distance is zero we have a sequence of gauge transformations gi so that

kui · A1 − A0k 2 → 0. L1

Writing A1 = A0 + a and ai = gi · A1 − A0 we have

dA0 gi = (ai − a)gi − gi a.

By bootstrapping using the fact that gi is uniformly bounded in ∞ 2 L we see that gi is bounded in L2. Thus the sequence of gauge transformations has a subsequence which converges in 2 2 ∞ the weak L2-topology, the strong L1-topology and in the L 2 weak-∗ topology. The strong L1 convergence implies that the limit still satisfies gg∗ = 1, a.e. and det g = 1, a.e. 2 Thus the limit is a gauge transformation g ∈ L2 carrying A1 to Tom Mrowka 18.937 2 A0. If A1 and A0 are both in some more regular space eg. Ls then g is in Gs+1.  Definition

Define BˆLn to be the completion of Bs with respect to dˆ.

In fact one can show that BLn is naturally identified with the ∞ quotient space of An/2−1 by Gn/2 but it seems easier to the ˆ author to think of Bn/2−1 as the completion. 2 and its L1 completion ˆ 2 ∗ ∗ SA() = {A+a|a ∈ L (X; T X⊗sl(E)), d a = 0 and kak 2 < }. 1,A A L1,A

Tom Mrowka 18.937 To state the result in general we need to discuss how to deal with reducible connections. If A is reducible StabA, the stabilizer of A, is a Lie subgroup of SU(n). StabA acts on SA and freely on Gs+1 and it is straightforward to see that the quotient

SA() × Gs+1/StabA is a smooth Hilbert manifold. The tangent space at the equivalence class of (A + a, 1) is identified with the quotient of

2 ∗ 2 {α|α ∈ Ls,A, dAα = 0} × Ls+1,A(X, su(E)) by the finite-dimensional (and hence closed) subspace

0 0 0 {(−[a, ξ ], ξ )|dAξ = 0}.

Tom Mrowka 18.937 Let λ1(A) denote the first positive eigenvalue of the Laplacian

∗ 2 ⊥ 2 ⊥ dAdA : L1 ∩ ker(dA) → L−1 ∩ ker(dA) and let κ = κ(A) be the norm of the Sobolev embedding

2 4 L1,A ,→ L .

Tom Mrowka 18.937 Proposition

For all A ∈ As if  > 0 satifies

−1 1 κ(A)(1 + λ1(A) ) 2 < 1/2 then the map

m : SA() × Gs+1/StabA → As −1 −1 m(A + a, u) = A + uau − (dAu)u is a Gs+1-equivariant diffeomorphism onto its image. The image 2 of m contains an open L1,A-ball about A in As.

Tom Mrowka 18.937 Proof. 0 1 0 Write ξ = ξ + ξ for the Hodge decomposition where dAξ = 0 1 2 and ξ is L -orthogonal to elements of the kernel of dA. First we check that the differential of m is injective if

−1 1 κ(A)(1 + λ1(A) ) 2 kak 2 < 1. L1,A

Since m is Gs-equivariant it suffices to check this at the equivalence classes of (A + a, 1). The differential is given by the map D(A+a,1)m(α, ξ) = dAξ + [ξ, a] + α

Tom Mrowka 18.937 Suppose that (α, ξ) ∈ ker(D(A+a,1)m) and is orthogonal to 0 0 0 {(−[a, ξ ], ξ )|dAξ = 0}. 1 2 1 so that in particular ξ = ξ . Take the L -inner product of dAξ with D(A+a,1)m(α, ξ) to get 2 n−1 1 kdAξk + (−1) h∗[∗dAξ, a], ξ i = 0. so we have 1 2 1 kdAξ kL2 ≤ kdAξkL2 kakL4 kξ kL4 1 1 ≤ κkdAξ kL2 kak 2 kξ kL2 L1,A 1,A 1 −1 2 1 2 ≤ κ(1 + λ(A) ) kdAξ kL2 kak 2 . L1,A Thus if −1 1 κn(1 + λ1(A) ) 2 kak 2 < 1. L1,A we have 1 dAξ = 0

Tom Mrowka 18.937 If this is the case we also have α = −[ξ, a], thus the differential of the induced map on SA() × Gs+1/Stab(A) is injective as required. The differential is a smoothly varying family of Fredholm maps which is invertible at the equivalence class of (1, A) and hence is invertible whenever it is injective. Thus we see that m is local diffeomorphism onto its image by the inverse function theorem. Note however that we can only conclude that Dm is invertible 2 but only uniformly invertible on Ls,A bounded sets.

Tom Mrowka 18.937 m is injective. It suffices to show that if g · (A + a) = A + b is in the slice then g ∈ StabA. The condition that g · (A + a) = A + b is equivalent to dAg = ga − bg. (14) ∗ Taking dA of this equation gives, using the slice condition ∗ ∗ dAa = dAb = 0,

∗ dAdAg = − ∗ dAg ∧ ∗a − ∗b ∧ ∗dAg. (15)

Use the Hodge decomposition g = g0 + g1 again and take the inner product of this equation with g1.

1 2 1 1 kdAg k 2 ≤ κ(kak 2 + kbk 2 )kdAg kL2 kg kL4 L L1,A L1,A −1 1/2 1 2 ≤ κ(1 + λ (A)) (kak 2 + kbk 2 )kdAg k 2 1 L1,A L1,A L

−1 1/2 1 So for κ(1 + λ1 (A)) < 1/2 we have u = 0 and hence u = u0 is in Stab(A) as required.

Tom Mrowka 18.937 It remains to check that the image of m contains a 2 neighborhood of A in the L1-topology. The argument is a variation on the Uhlenbeck’s connectedness argument. Let

2 Uδ = {A + a|a ∈ Ls,A(X; su(E)), kak 2 < δ} . L1,A ˆ By Lemma 31 Uδ contains and is contained in a d-ball about A. Uδ is a connected open set so to prove that Uδ ⊂ m(Gn/2+1 × SA()) it is enough to show that Uδ ∩ m(Gn/2+1 × SA()) is a closed subset of Uδ. Then the intersection is a component of Uδ. Take a sequence m(ui , A + ai ) = A + bi contained in the intersection and 2 converging in the Ls topology to A∞ ∈ Uδ. We need the following lemma.

Tom Mrowka 18.937 Lemma

For all A ∈ As there is a constants K > 0 and δ1 so that for all A + a, A + b ∈ As and g ∈ Gs+1 with g · (A + a) = A + b and kak 2 + kbk 2 ≤ δ1 we have L1,A L1,A

kdAgk 2 + kak 2 ≤ K kbk 2 . L1,A L1,A L1,A The constant K , as will be clear from the proof below, depends 2 upper semi-continuously on A in Ls topology. It is not clear if 2 this holds in the L1-topology.

Tom Mrowka 18.937 Proof. The gauge transformation g satisfies the equation (14), ∗ and as above. Applying dA to this equation now gives ∗ ∗ dAdAg = − ∗ dAg ∧ ∗a + ∗b ∧ ∗dAg − dAbg. (16) ∗ 2 2 For and A ∈ A1 we have that the operator dAdA : LA,2 → L is Fredholm, self-adjoint and varies continuously as a function of 2 A ∈ L3/2,A. We can then find a constant κ depending on A ∈ As so that ∗ kd ukLn ≤ κkd d uk n . (17) A A A LA,−1 Combining this with equation (16) yields
kdAukLn ≤ κ ζ(kakLn +kbkLn )kdAukLn +ζkbkLn +kbkLn kdAukLn . (18) Here we have used the multiplication n n n L × L → LA,−1 n/2 n or equivalently the embedding L ,→ LA,−1 whose norm is the constant ζ and the inequality ∗ kd buk n ≤ ζkbkLn + kbkLn kd ukLn . A LA,−1 A Tom Mrowka 18.937 n/n−1 n/n−2 where ζ is the norm of the embedding LA,1 → L which is dual to the embedding above and hence has the same norm. n/n−1 To prove this inequality take w ∈ LA,1 with , the dual space of n LA,−1. Then we then have Z Z Z ∗ dAbuw = bdAuw + budAw X X X and hence Z ∗ | d buw| ≤ ζkbk n kd uk n kwk n/n−1 + kbk n kd wk n/n−1 A L A L L L A L X A,1 Thus if, say, κ ζkakLn + (1 + ζ)kbkLn ) < 1/2 we can rearrange this inequality to get the desired estimate on dAu:

kdAukLn ≤ 2κζkbkLn . (19)

To get the estimate on a we use the equation

−1 −1 a = u bu − (dAu )u.

Then, using |u| = 1 a.e. , we have obtain

kakLn ≤ kbkLn + kdAukLn .

Combining this with inequality (19) yields the claim. 

Tom Mrowka 18.937