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Tom Mrowka Fall Semester, 2011 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 2 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) 2 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 2 H∗(X) written as h = h− + h+ we can find DX± (h±) 2 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)jajjbj+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)jaj(jbj+jcj[b^[c^a]]+(−1)jcj(jaj+jcj[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 1 1 ∗ 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 g^ : P ! G so that g^(ph) = h−1g^(p)h via the identifcation g(p) = pg^(p) then g(ph) = phg^(ph) = phh−1g^(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 2 TpP let X denote the projection to HAjp. Then expression ~ h ~ h −A([X ; Y ])jp ~ ~ 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 )jp = −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 ξ^ = ιp(ξ); η^ = ιp(η) are vertical vectors. Then 1 0 = dA(ξ;^ η^) + [A ^ A](ξ;^ η^) 2 = ξ^(η) − η^(ξ) − A([ξ;^ η^]) + [A(ξ^); A(^η)] = 0 − 0 − [ξ; η] + [ξ; η]: If ξ^ = ιp(ξ) and Y is horizontal we have 1 0 = dA(X; Y ) + [A; A](X; Y ) 2 = ξ^(0) − Y (ξ) − A([ξ;^ 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) = Adg^ ◦ FA: 1 2 Thus we can view FA 2 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.
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