Monopoles and Relations Between Donaldson and Seiberg-Witten Invariants

Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Superconformal simple type and Witten’s formula Local and global gluing maps for SO(3) monopoles Bibliography

SO(3) monopoles and relations between Donaldson and Seiberg-Witten invariants

Paul Feehan and Thomas Leness

Rutgers University, New Brunswick & Florida International University

Wednesday, August 9, 2017

Low Dimensional Topology and Gauge Theory Banﬀ International Research Station Casa Matem´aticaOaxaca

1 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Superconformal simple type and Witten’s formula Local and global gluing maps for SO(3) monopoles Bibliography Outline

1 Introduction and main results

2 Review of Donaldson and Seiberg-Witten invariants

3 SO(3)-monopole cobordism

4 Superconformal simple type and Witten’s formula

5 Local and global gluing maps for SO(3) monopoles

6 Bibliography

2 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography

Introduction and main results

3 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography IntroductionI

In his article [52], Witten (1994) Gave a formula expressing the Donaldson series in terms of Seiberg-Witten invariants for standard four-manifolds, Outlined an argument based on supersymmetric quantum ﬁeld theory, his previous work [51] on topological quantum ﬁeld theories (TQFT), and his work with Seiberg [44, 45] explaining how to derive this formula. (We call a four-manifold standard if it is closed, connected, + oriented, and smooth with odd b ≥ 3 and b1 = 0.) In a later article [35], Moore and Witten extended the scope of Witten’s previous formula by allowing + four-dimensional manifolds with b1 > 0 and b = 1, and

4 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography IntroductionII

provided more details underlying the derivation of these formulae using supersymmetric quantum ﬁeld theory.

Using similar supersymmetric quantum field theoretic ideas methods, Marinõ,Moore, and Peradze (1999) also showed that a certain low-degree polynomial part of the Donaldson series always vanishes [31, 32], a consequence of their notion of superconformal simple type. Marinõ,Moore, and Peradze noted that this vanishing would confirm a conjecture (attributed to Fintushel and Stern) for a lower bound on the number of (Seiberg-Witten) basic classes of a four-dimensional manifold.

5 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography IntroductionIII

Soon after the Seiberg-Witten invariants were discovered, Pidstrigatch and Tyurin (1994) proposed a method [42] to prove Witten’s formula using a classical field theory paradigm via the space of SO(3) monopoles which simultaneously extend the Anti-self-dual SO(3) connections, defining Donaldson invariants, and U(1) monopoles, defining Seiberg-Witten invariants.

6 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography IntroductionIV

The Pidstrigatch-Tyurin SO(3)-monopole paradigm is intuitively appealing, but there are also signiﬁcant technical diﬃculties in such an approach. In this lecture, we summarize some ideas in our proofs that, for all standard four-manifolds, the SO(3)-monopole paradigm and

Seiberg-Witten simple type =⇒ Superconformal simple type, Superconformal simple type =⇒ Witten’s Conjecture.

Taken together, these implications prove Marin˜o,Moore, and Peradze’s Conjecture on superconformal simple type and Fintushel and Stern’s Conjecture on the lower bound on the number of basic classes, and

7 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography IntroductionV

Witten’s Conjecture on the relation between Donaldson and Seiberg-Witten invariants. It is unknown whether all four-manifolds have Seiberg-Witten simple type. More details can be found in 1 P. M. N. Feehan and T. G. Leness, Gluing maps for SO(3) monopoles and invariants of smooth four-manifolds, in preparation; based on recent work and arXiv:math/9812060 and arXiv:math/9907107. and our two articles 1 P. M. N. Feehan and T. G. Leness, The SO(3) monopole cobordism and superconformal simple type, arXiv:1408.5307.

8 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Introduction VI

2 P. M. N. Feehan and T. G. Leness, Superconformal simple type and Witten’s conjecture, arXiv:1408.5085. These are in turn based on methods described earlier in our 1 P. M. N. Feehan and T. G. Leness, A general SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants, Memoirs of the American Mathematical Society, in press, arXiv:math/0203047. 2 P. M. N. Feehan and T. G. Leness, Witten’s conjecture for many four-manifolds of simple type, Journal of the European Mathematical Society 17 (2015), 899–923. Additional useful references include

9 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Introduction VII

1 P. M. N. Feehan and T. G. Leness, PU(2) monopoles. I: Regularity, compactness and transversality, Journal of Differential Geometry 49 (1998), 265–410. 2 P. M. N. Feehan, Generic metrics, irreducible rank-one PU(2) monopoles, and transversality, Communications in Analysis & Geometry 8 (2000), 905–967. 3 P. M. N. Feehan and T. G. Leness, PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces, Journal fr die Reine und AngewändteMathematik 538 (2001), 57–133. 4 P. M. N. Feehan and T. G. Leness, PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten’s conjecture in low degrees, Journal fürdie Reine und AngewändteMathematik 538 (2001), 135–212.

10 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Introduction VIII

5 P. M. N. Feehan and T. G. Leness, SO(3) monopoles, level-one Seiberg-Witten moduli spaces, and Witten’s conjecture in low degrees, Topology and its Applications 124 (2002), 221–326.

11 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsI

A closed, oriented four-manifold X has an intersection form,

QX : H2(X ; Z) × H2(X ; Z) → Z.

One lets b±(X ) denote the dimensions of the maximal positive or negative subspaces of the form QX on H2(X ; Z) and

4 X i + − e(X ) := (−1) bi (X ) and σ(X ) := b (X ) − b (X ) i=0 denote the Euler characteristic and signature of X , respectively.

12 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsII

We deﬁne the characteristic numbers,

2 c1 (X ) := 2e(X ) + 3σ(X ), (1) χh(X ) := (e(X ) + σ(X ))/4, 2 c(X ) := χh(X ) − c1 (X ). Recall that a four-manifold X is standard if it is closed, connected, + oriented, and smooth with odd b (X ) ≥ 3 and b1(X ) = 0. + (The methods we will describe allow b (X ) = 1 and b1(X ) > 0.) For a standard four-manifold, the Seiberg-Witten invariants comprise a function,

c SWX : Spin (X ) → Z,

13 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsIII

on the set of spinc structures on X . The set of Seiberg-Witten basic classes, B(X ), is the image under c 2 c1 : Spin (X ) → H (X ; Z) of the support of SWX , that is

2 B(X ) := {K ∈ H (X ; Z): K = c1(s) with SWX (s) 6= 0},

and is ﬁnite. 2 2 X has Seiberg-Witten simple type if K = c1 (X ), ∀ K ∈ B(X ). 2 2 (Here, c1(s) = c1 (X ) ⇐⇒ the moduli space, Ms, of Seiberg-Witten monopoles has dimension zero.)

14 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsIV

According to Kronheimer and Mrowka [26, Theorem 1.7 (a)], the Donaldson series of a standard four-manifold of simple type (in 2 their sense), for any w ∈ H (X ; Z), is given by

2 w QX (h)/2 X (w +K·w)/2 hK,hi (2) DX (h) = e (−1) βX (K)e , K∈H2(X ;Z)

as an equality of analytic functions of h ∈ H2(X ; R), where

2 (3) βX : H (X ; Z) → Q,

is a function such that βX (K) 6= 0 for at most ﬁnitely many 2 classes, K, which are integral lifts of w2(X ) ∈ H (X ; Z/2Z) (the Kronheimer-Mrowka basic classes).

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Conjecture 1.1 (Witten’s Conjecture)

Let X be a standard four-manifold. If X has Seiberg-Witten simple type, then X has Kronheimer-Mrowka simple type, the Seiberg-Witten and Kronheimer-Mrowka basic classes coincide, 2 and for any w ∈ H (X ; Z),

2 w 2−(χh−c1 ) QX (h)/2 (4) DX (h) = 2 e 1 2 X (w +c1(s)·w) hc1(s),hi × (−1) 2 SWX (s)e , s∈Spinc (X )

∀ h ∈ H2(X ; R).

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As deﬁned by Mari˜no,Moore, and Peradze, [32, 31], a manifold X has superconformal simple type if c(X ) ≤ 3 or c(X ) ≥ 4 and for 2 w ∈ H (X ; Z) characteristic,

w,i (5) SWX (h) = 0 for i ≤ c(X ) − 4

and all h ∈ H2(X ; R), where

w,i X 1 2 2 (w +c1(s)·w) i SWX (h) := (−1) SWX (s)hc1(s), hi s∈Spinc (X )

From [8], we have the

17 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsVII Theorem 1.2 (All standard four-manifolds with Seiberg-Witten simple type have superconformal simple type)

(See F and Leness [8, Theorem 1.1].) If X is a standard four-manifold of Seiberg-Witten simple type, then X has superconformal simple type.

Marin˜o,Moore, and Peradze had previously shown [32, Theorem 8.1.1] that if the set of Seiberg-Witten basic classes, B(X ), is non-empty and X has superconformal simple type, then

(6) |B(X )/{±1}| ≥ [c(X )/2].

18 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsVIII For example, suppose X is the K3 surface. Because c(X ) ≤ 3, the K3 surface is superconformal simple type by our deﬁnition. It is known that B(X ) = {0}, so |B(X )/{±1}| = 1, while

+ − b1(X ) = 0, b (X ) = 3, b (X ) = 19,

2 which gives e(X ) = 24, σ(X ) = −16, c1 (X ) = 0, χh(X ) = 2, and c(X ) = 2, so [c(X )/2] = 1 and equality holds in (6).

19 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Statements of main resultsIX

Theorem 1.2 and [32, Theorem 8.1.1] therefore yield a proof of the following result, ﬁrst conjectured by Fintushel and Stern [15].

Corollary 1.3 (Lower bound for the number of basic classes)

(See F and Leness [8, Corollary 1.2]) Let X be a standard four-manifold of Seiberg-Witten simple type. If B(X ) is non-empty and c(X ) ≥ 3, then the number of basic classes obeys the lower bound (6).

From [10], we have the

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Theorem 1.4 (Superconformal simple type =⇒ Witten’s Conjecture holds for all standard four-manifolds)

(See F and Leness [10, Theorem 1.2].) If a standard four-manifold with has superconformal simple type, then it satisﬁes Witten’s Conjecture 1.1.

Combining Theorems 1.2 and 1.4 thus yields the following

Corollary 1.5 (Witten’s Conjecture holds for all standard four-manifolds)

(See F and Leness [10, Corollary 1.3] or [8, Corollary 1.4].) If X is a standard four-manifold of Seiberg-Witten simple type then X satisﬁes Witten’s Conjecture 1.1.

21 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Introduction SO(3)-monopole cobordism Statements of main results Superconformal simple type and Witten’s formula Further results and future directions Local and global gluing maps for SO(3) monopoles Bibliography Further results and future directionsI

Kronheimer and Mrowka applied our SO(3)-monopole cobordism formula (Theorem 3.3) to give their ﬁrst of two proofs of Property P for knots in [27]. Property P asserts that +1 surgery on a non-trivial knot K in S3 yields a three-manifold which is not a homotopy sphere. In parallel to their role in conﬁrming the relationship between the Donaldson and Seiberg-Witten invariants of four-manifolds, one could use SO(3) monopoles to explore the relationship between the instanton (Yang-Mills) and monopole (Seiberg-Witten) Floer homologies of three-manifolds.

22 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography

Review of Donaldson and Seiberg-Witten invariants

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Seiberg-Witten invariants

24 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsI

Detailed expositions of the theory of Seiberg-Witten invariants, introduced by Witten in [52], are provided in [28, 36, 39]. These invariants deﬁne an integer-valued map with ﬁnite support,

c SWX : Spin (X ) → Z,

on the set of spinc structures on X .

c ± Aside:A spin structure, s = (W , ρW ) on X , consists of a pair of complex rank-two bundles W ± → X and a Cliﬀord multiplication ∗ ± ∓ map ρ = ρW : T X → HomC(W , W ) such that [25, 29, 43]

∗ ∗ (7) ρ(α) = −ρ(α) and ρ(α) ρ(α) = g(α, α) idW ,

25 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsII

for all α ∈ C ∞(T ∗X ), where W = W + ⊕ W − and g denotes the Riemannian metric on T ∗X . The Cliﬀord multiplication ρ induces canonical isomorphisms Λ± =∼ su(W ±), where Λ± = Λ±(T ∗X ) are the bundles of self-dual and anti-self-dual two-forms, with respect to the Riemannian metric g on T ∗X . 1 Any two spin connections on W diﬀer by an element of Ω (X ; iR), since the induced connection on su(W ) =∼ Λ2 is determined by the Levi-Civita connection for the metric g on T ∗X . Consider a spin connection, B, on W and section Ψ ∈ C ∞(W +).

26 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsIII We call a pair (B, Ψ) a Seiberg-Witten monopole if

+ −1 ∗ Tr(F ) − τρ (Ψ ⊗ Ψ )0 − η = 0, (8) B DB Ψ + ρ(ϑ)Ψ = 0,

+ + where, writing u(W ) = iR ⊕ su(W ), + ∞ + + FB ∈ C (Λ ⊗ u(W )) is the self-dual component of the curvature FB of B, and + ∞ + + Tr(FB ) ∈ C (Λ ⊗ iR) is the trace part of FB , ∞ + ∞ − DB = ρ ◦ ∇B : C (W ) → C (W ) is the Dirac operator deﬁned by the spin connection B, The perturbation terms τ and ϑ are as in our version of the forthcoming SO(3)-monopole equations (17),

27 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsIV

η ∈ C ∞(iΛ+) is an additional perturbation term, The quadratic term Ψ ⊗ Ψ∗ lies in C ∞(iu(W +)) and ∗ (Ψ ⊗ Ψ )0 denotes the traceless component lying in ∞ + −1 ∗ ∞ + C (isu(W )), so ρ (Ψ ⊗ Ψ )0 ∈ C (iΛ ). In the usual presentation of the Seiberg-Witten equations, one takes τ = idΛ+ and ϑ = 0, while η is a generic perturbation. However, in order to identify solutions to the Seiberg-Witten equations (8) with reducible solutions to the forthcoming SO(3)-monopole equations (17), one needs to employ the perturbations given in equation (8) and choose

(9) η = F + , AΛ

28 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsV

where AΛ is the ﬁxed unitary connection on the line bundle 1 + 2 det 2 (V ) with Chern class denoted by c1(t) = Λ ∈ H (X ; Z) and

represented by the real two-form (1/2πi)FAΛ , where V = W ⊗ E and V ± = W ± ⊗ E. Here, E is a rank-two, Hermitian bundle over X arising in deﬁnitions of anti-self-dual SO(3) connections and SO(3) monopoles. c Given a spin structure, s, one may construct a moduli space, Ms, of solutions to the Seiberg-Witten monopole equations, modulo gauge equivalence.

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The space, Ms, is a compact, ﬁnite-dimensional, oriented, smooth manifold (for generic perturbations of the Seiberg-Witten monopole equations) of dimension

1 (10) dim M = c (s)2 − 2χ − 3σ , s 4 1 and contains no zero-section points [B, 0].

When Ms has odd dimension, the Seiberg-Witten invariant, SWX (s), is deﬁned to be zero.

When Ms has dimension zero, then SWX (s), is deﬁned by counting the number of points in Ms.

30 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsVII

When Ms has even positive dimension ds , one deﬁnes

ds /2 SWX (s) := hµs , [Ms]i,

where µs = c1(Ls) is the ﬁrst Chern class of the universal complex line bundle over the conﬁguration space of pairs. c + 2 If s ∈ Spin (X ), then c1(s) := c1(W ) ∈ H (X ; Z) and 2 c1(s) ≡ w2(X ) (mod 2) ∈ H (X ; Z/2Z), where w2(X ) is second Stiefel-Whitney class of X .

One calls c1(s) a Seiberg-Witten basic class if SWX (s) 6= 0. Deﬁne

(11) B(X ) = {c1(s): SWX (s) 6= 0}.

31 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Seiberg-Witten invariantsVIII 2 c 2 If H (X ; Z) has 2-torsion, then c1 : Spin (X ) → H (X ; Z) is not injective and as we will work with functions involving real homology and cohomology, we deﬁne

0 2 X (12) SWX : H (X ; Z) 3 K 7→ SWX (s) ∈ Z. −1 s∈c1 (K) With the preceding deﬁnition, Witten’s Formula (4) is equivalent to

2 w 2−(χh−c1 ) QX (h)/2 (13) DX (h) = 2 e X 1 (w 2+K·w) 0 hK,hi × (−1) 2 SWX (K)e . K∈B(X )

A four-manifold, X , has Seiberg-Witten simple type if SWX (s) 6= 0 2 2 implies that c1 (s) = c1 (X ) (or, in other words, dim Ms = 0). 32 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography

Donaldson invariants

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2 For w ∈ H (X ; Z), the Donaldson invariant is a linear function,

w DX : A(X ) → R,

where A(X ) = Sym(Heven(X ; R)), the symmetric algebra.

For h ∈ H2(X ; R) and a generator x ∈ H0(X ; Z), one deﬁnes w δ−2m m DX (h x ) = 0 unless

2 (14) δ ≡ −w − 3χh(X ) (mod 4).

w δ−2m m Given (14), then DX (h x ) is (heuristically) deﬁned by pairing 1 A cohomology class µ(z) of dimension 2δ on the conﬁguration space of SO(3) connections on su(E), corresponding to the δ−2m m degree-δ element z = h x ∈ A(X ), and 34 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Donaldson invariantsII

¯ w 2 A fundamental class [Mκ (X )] defined by the Uhlenbeck w compactification of a moduli space Mκ (X ) of anti-self-dual 1 SO(3) connections on su(E), where κ = − 4 p1(su(E)) and E is a rank-two Hermitian bundle with w = c1(E). See Donaldson [2], Donaldson and Kronheimer [3], Friedman and Morgan [16], Kronheimer and Mrowka [26], and Morgan and w δ−2m m Mrowka [37] for precise definitions of DX (h x ). Suppose A is a unitary connection on a Hermitian vector bundle E over X and Aˆ is the induced SO(3) connection on su(E). One calls Aˆ anti-self-dual (with respect to the metric, g, on X ) if

F + = 0, Aˆ

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where F is the curvature of Aˆ and F + is its self-dual component Aˆ Aˆ with respect to the splitting, Λ2(T ∗X ) = Λ+ ⊕ Λ−. 1 w We denote κ = − 4 p1(su(E)) and w = c1(E) and write Mκ (X ) for the moduli space of gauge-equivalence classes of anti-self-dual SO(3) connections on su(E). A four-manifold has Kronheimer-Mrowka simple type if for all 2 w ∈ H (X ; Z) and all z ∈ A(X ) one has

w 2 w (15) DX (x z) = 4DX (z).

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This equality implies that the Donaldson invariants are determined by the Donaldson series (see Kronheimer and Mrowka [26, Section 2]), the formal power series

w w 1 h (16) DX (h) := DX ((1 + 2 x)e ), ∀ h ∈ H2(X ; R), and computed using their formula (2). More generally (see Kronheimer and Mrowka [24]), a four-manifold X has ﬁnite type or type τ if

w 2 τ DX ((x − 4) z) = 0,

for some τ ∈ N and all z ∈ A(X ).

37 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Seiberg-Witten invariants Superconformal simple type and Witten’s formula Donaldson invariants Local and global gluing maps for SO(3) monopoles Bibliography Donaldson invariantsV

Kronheimer and Mrowka conjectured [24] that all standard four-manifolds X have finite type and state an analogous formula w for the series DX (h). Proofs of different parts of their conjecture have been given by Frøyshov [17, Corollary 1], Muñoz[40, Corollary 7.2 & Proposition 7.6], and Wieczorek [50, Theorem 1.3].

38 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography

SO(3)-monopole cobordism

39 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsI

The SO(3)-monopole equations take the form,

−1 + −1 ∗ ad (F ) − τρ (Φ ⊗ Φ )00 = 0, (17) Aˆ DAΦ + ρ(ϑ)Φ = 0.

where A is a spin connection on V = W ⊗ E and E is a Hermitian, rank-two bundle, Φ ∈ C ∞(W + ⊗ E), F + ∈ C ∞(Λ+ ⊗ so(su(E))) is the self-dual component of the Aˆ ˆ curvature FAˆ of the induced SO(3) connection, A, on su(E), ad−1(F +) ∈ C ∞(Λ+ ⊗ su(E)), Aˆ

40 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsII ∞ + ∞ − DA = ρ ◦ ∇A : C (V ) → C (V ) is the Dirac operator, ∞ + ∞ 1 τ ∈ C (GL(Λ )) and ϑ ∈ C (Λ ⊗ C) are perturbation parameters. Aside: For Φ ∈ C ∞(V +), we let Φ∗ denote its pointwise Hermitian ∗ dual and let (Φ ⊗ Φ )00 be the component of Φ ⊗ Φ∗ ∈ C ∞(iu(V +)) which lies in the factor su(W +) ⊗ su(E) of the decomposition,

+ ∼ + iu(V ) = R ⊕ isu(V ).

The Cliﬀord multiplication ρ deﬁnes an isomorphism ρ :Λ+ → su(W +) and thus an isomorphism

+ ∼ + ρ = ρ ⊗ idsu(E) :Λ ⊗ su(E) = su(W ) ⊗ su(E). 41 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsIII

Note also that ad : su(E) → so(su(E)) is an isomorphism of real vector bundles.

We ﬁx, once and for all, a smooth, unitary connection AΛ on the 1 + square-root determinant line bundle, det 2 (V ), and require that our unitary connections A on V = V + ⊕ V − induce the resulting unitary connection on det(V +),

det + (18) A = 2AΛ on det(V ),

det + where A is the connection on det(V ) induced by A|V + .

42 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsIV

det If a unitary connection A on V induces a connection A = 2AΛ + 1 + on det(V ), then it induces the connection AΛ on det 2 (V ).

We let Mt denote the moduli space of solutions to the SO(3)-monopole equations (17) moduli gauge-equivalence, where t = (ρ, W ±, E).

The moduli space, Mt, of SO(3) monopoles contains the w Moduli subspace of anti-self-dual SO(3) connections, Mκ , identiﬁed with the subset of equivalence classes of SO(3) monopoles, [A, 0], withΦ ≡ 0, and

43 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsV

Moduli subspaces, Ms, of Seiberg-Witten monopoles, identiﬁed with subsets of equivalence classes of SO(3) monopoles, [A1 ⊕ A2, Φ1 ⊕ 0], where the connections, A on E, become reducible with respect to diﬀerent splittings, E = L1 ⊕ L2, and Ai is a U(1) connection on Li , and ± s = (ρ, W ⊕ L1).

∗,0 We let Mt denote the complement in Mt of these subspaces of zero-section and reducible SO(3) monopoles. ∗,0 For generic perturbations, Mt is a (ﬁnite-dimensional) smooth, orientable manifold (see F [4], F and Leness [11], or Teleman [49]).

44 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsVI

± ± For t = (W ⊗ E, ρ) and ` ≥ 0, deﬁne t(`) := (W ⊗ E`, ρ), where

c1(E`) = c1(E), c2(E`) = c2(E) − `.

We deﬁne the space of ideal SO(3) monopoles by

N N G ` I Mt := Mt(`) × Sym (X ) , `=0

where Sym`(X ) is the symmetric product of X and N is a large integer (depending on perturbations and geometry of X and E).

The Uhlenbeck compactiﬁcation, M¯t, is the closure of Mt in N I Mt (see F and Leness [11]).

45 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole equationsVII

Because dim Mt(`) = dim Mt − 6`, one has

` dim Mt(`) × Sym (X ) = dim Mt − 2`.

For each level, `, in the range 0 ≤ ` ≤ N, the top( ` = 0) and lower level( ` ≥ 1) subspaces of M¯t,

¯ ` Mt ∩ Mt(`) × Sym (X ) , 0 ≤ ` ≤ N,

may contain (ideal) Seiberg-Witten moduli subspaces,

` ` Ms × Sym (X ) ⊂ Mt(`) × Sym (X ).

46 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsI

Our forthcoming SO(3)-monopole cobordism formula (20) is proved by evaluating the pairings of cup products of suitable cohomology classes on M¯t with (or intersecting geometric representatives of those classes with) the asd 1 Link Lt of the moduli subspace of anti-self-dual SO(3) ¯ w connections, Mκ , potentially giving multiples of the Donaldson invariant,

2 Links Lt,s of the Seiberg-Witten moduli subspaces, ` Ms × Sym (X ), giving sums of multiples of the Seiberg-Witten invariants.

47 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsII

The following ﬁgure illustrates the SO(3)-monopole cobordism ¯ 1 ¯ w between codimension-one links in Mt/S of Mκ and ` Msi × Sym (X ).

48 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsIII

49 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsIV

We shall use a definition of a link of a stratum in smoothly stratified space, following Mather [33] and Goresky-MacPherson [19]. We need only consider the relatively simple case of a stratified space with two strata since the lower strata in

∼ ∗,0 w G (19) Mt = Mt t Mκ ∪ Ms s

do not intersect when Mt contains no reducible, zero-section solutions.

50 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsV

The finite union in (19) over s is over the subset of all c spin structures for which Ms is non-empty and for which there is a splitting t = s ⊕ s0. Aside: The space, Z, in the forthcoming Definition 3.1 is a smoothly stratified space (with two strata) in the sense of Morgan, Mrowka, and Ruberman [38, Chapter 11].)

51 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsVI

Deﬁnition 3.1 (Link of a stratum in a smoothly stratiﬁed space)

Let Z be a closed subset of a smooth, Riemannian manifold M, and suppose that Z = Z0 ∪ Z1, where Z0 and Z1 are locally closed, smooth submanifolds of M and Z1 ⊂ Z¯0. 0 Let NZ1 be the normal bundle of Z1 ⊂ M and let O ⊂ NZ1 be an open

neighborhood of the zero section Z1 ⊂ NZ1 such that there is a

diﬀeomorphism γ, commuting with the zero section of NZ1 (so 0 0 γ|Z1 = idZ1 ), from O onto an open neighborhood γ(O ) of Z1 ⊂ M. 0 Let O b O be an open neighborhood of the zero section Z1 ⊂ NZ1 , where O¯ = O ∪ ∂O ⊂ O0 is a smooth manifold-with-boundary.

Then LZ1 := Z0 ∩ γ(∂O) is a link of Z1 in Z0.

52 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsVII

¯∗,0 1 The compactification Mt /S defines a compact cobordism, stratified by smooth oriented manifolds, between

asd G Lt and Lt,s. s∈Red(t)

1 asd For δ + ηc = 2 dim Lt , this cobordism gives the following equality (see F and Leness [6, Equation (1.6.1)]),

¯ δ−2m m ¯ηc asd # V (h x ) ∩ W ∩ Lt X 1 (w 2−σ)+ 1 (w 2+(w−c (t))·c (s)) = − (−1) 2 2 1 1 (20) s∈Red(t)

¯ δ−2m m ¯ηc × # V (h x ) ∩ W ∩ Lt,s .

53 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism formula for link pairingsVIII

The left-hand side of (20) is obtained by computing the 1 intersection number for geometric representatives on M¯t/S with ¯ w the link of the moduli subspace Mκ of anti-self-dual SO(3) connections. The right-hand side of (20) is obtained by computing the 1 intersection numbers for geometric representatives on M¯t/S with ` the links of the moduli subspaces Ms × Sym (X ) of ideal 1 Seiberg-Witten monopoles appearing in M¯t/S .

54 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsI

It will be more convenient to have Witten’s Formula (4) expressed at the level of the Donaldson polynomial invariants rather than the Donaldson power series which they form. Let B0(X ) be a fundamental domain for the action of {±1} on B(X ).

55 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsII

Lemma 3.2 (Donaldson invariants implied by Witten’s formula)

(See F and Leness [12, Lemma 4.2].) Let X be a standard four-manifold. Then X satisﬁes equation (4) and has Kronheimer-Mrowka simple type if w δ−2m m and only if the Donaldson invariants of X satisfy DX (h x ) = 0 for 2 2 δ 6≡ −w − 3χh (mod 4) and for δ ≡ −w − 3χh (mod 4) satisfy

w δ−2m m X X ε(w,K) (21) DX (h x ) = (−1) ν(K) i+2k K∈B0(X ) =δ−2m SW 0 (K)(δ − 2m)! × X hK, hii Q (h)k , 2k+c(X )−3−mk!i! X

56 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsIII

Lemma 3.2 (Donaldson invariants implied by Witten’s formula) where 1 (22) ε(w, K) := (w 2 + w · K), 2 and ( 1 if K = 0, (23) ν(K) = 2 1 if K 6= 0.

The SO(3)-monopole cobordism formula given below provides an expression for the Donaldson invariant in terms of the Seiberg-Witten invariants.

57 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsIV

Theorem 3.3 (Cobordism formula for Donaldson invariants) (See F and Leness [6, Main Theorem].) Let X be a standard four-manifold of Seiberg-Witten simple type. Assume further that w, Λ ∈ H2(X ; Z) and δ, m ∈ N satisfy

(24a) w − Λ ≡ w2(X ) (mod 2), 2 (24b) I (Λ) = Λ + c(X ) + 4χh(X ) > δ, 2 (24c) δ ≡ −w − 3χh(X ) (mod 4), (24d) δ − 2m ≥ 0.

Then, for any h ∈ H2(X ; R) and positive generator x ∈ H0(X ; Z),

58 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsV

Theorem 3.3 (Cobordism formula for Donaldson invariants)

X 1 2 1 2 w δ−2m m 2 (w −σ)+ 2 (w +(w−Λ)·K) 0 (25) DX (h x ) = (−1) SWX (K) K∈B(X ) 2 × fδ,m(χh(X ), c1 (X ), K, Λ)(h),

where the map,

2 2 fδ,m(h): Z × Z × H (X ; Z) × H (X ; Z) → R[h], takes values in the ring of polynomials in the variable h with

59 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsVI

Theorem 3.3 (Cobordism formula for Donaldson invariants) real coeﬃcients, is universal (independent of X ) and is given by

2 (26) fδ,m(χh(X ), c1 (X ), K, Λ)(h) X 2 2 i j k := ai,j,k (χh(X ), c1 (X ), K · Λ, Λ , m) hK, hi hΛ, hi QX (h) . i+j+2k =δ−2m

For each triple, i, j, k ∈ N, the coeﬃcients,

ai,j,k : Z × Z × Z × Z × N → R, are universal (independent of X ) real analytic functions of the variables 2 2 χh(X ), c1 (X ), c1(s) · Λ, Λ , and m.

60 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsVII

The left-hand side of the SO(3)-monopole cobordism formula (25) is obtained by computing the intersection number for geometric 1 representatives on M¯t/S with the link of the moduli subspace ¯ w Mκ of anti-self-dual SO(3) connections. ¯ w One uses the ﬁber-bundle structure of the link over Mκ to compute the intersection number and show that this is equal to a w δ−2m m multiple of the Donaldson invariant, DX (h x ). The right-hand side of the SO(3)-monopole cobordism formula (25) is obtained by computing the intersection numbers for 1 geometric representatives on M¯t/S with the links of the moduli ` subspaces Ms × Sym (X ) of ideal Seiberg-Witten monopoles 1 appearing in M¯t/S .

61 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and Donaldson invariantsVIII

One uses the ﬁber-bundle structure of the link over each ` Seiberg-Witten moduli space, Ms × Sym (X ), to compute the intersection number and show that this is equal to a multiple of a 0 2 Seiberg-Witten invariant, SWX (K), for each K ∈ H (X ; Z) with c1(s) = K.

62 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and algebraic geometryI

Aside: When X is a complex projective surface, T. Mochizuki [34] proved a formula (see Göttsche,Nakajima, and Yoshioka [21, Theorem 4.1]) expressing the Donaldson invariants in a form similar to our SO(3)-monopole cobordism formula (Theorem 3.3). The coefficients in Mochizuki’s formula are given as the residues of ∗ a generating function for integrals of C -equivariant cohomology classes over the product of Hilbert schemes of points on X . In [21, p. 309], Göttsche,Nakajima, and Yoshioka conjecture that the coefficients in Mochizuki’s formula (which are meaningful for any standard four-manifold) and in our SO(3)-monopole cobordism formula are the same.

63 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and algebraic geometryII

G¨ottsche,Nakajima, and Yoshioka prove an explicit formula for complex projective surfaces relating Donaldson invariants and Seiberg-Witten invariants of standard four-manifolds of Seiberg-Witten simple type using an instanton-counting formula due to Nekrasov and verify Witten’s Conjecture for complex projective surfaces. In [21, p. 323], G¨ottsche,Nakajima, and Yoshioka discuss the relationship between their approach, Mochizuki’s formula, and our SO(3)-monopole cobordism formula. See also [20, pp. 344–347] for a related discussion concerning their wall-crossing formula for the Donaldson invariants of a four-manifold with b+ = 1.

64 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsI

In addition to its application to deriving a formula for Donaldson invariants, the SO(3)-monopole cobordism may also be used to derive relations among Seiberg-Witten invariants, as in the forthcoming Theorem 3.4.

65 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsII

Theorem 3.4 (SO(3)-monopole cobordism formula vanishing)

(See F and Leness [8, Theorem 3.3].) Let X be a standard four-manifold of Seiberg-Witten simple type. Assume that m, n ∈ N satisfy

(27a) n ≤ 2χh(X ), (27b) 1 < n, (27c) 0 ≤ c(X ) − n − 2m − 1.

We abbreviate the coeﬃcients in equation (25) in Theorem 3.3 by

2 (28) ai,0,k := ai,0,k (χh(X ), c1 (X ), 0, 0, m, 2χh(X ) − n).

66 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsIII

Theorem 3.4 (SO(3)-monopole cobordism formula vanishing)

2 Then, for A = c(X ) − n − 2m − 1 and w ∈ H (X ; Z) characteristic,

2χh(X )−n X w,A+2k 2χh(X )−n+k (29) 0 = aA+2k,0,2χh(X )−n−k SWX (h)QX (h) . k=0

To show that equation (29) is non-trivial, we now demonstrate, in a computation similar to that due to Kotschick and Morgan [23, Theorem 6.1.1], that the coeﬃcient of the term in (29) including the highest power of QX is non-zero.

67 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsIV

Proposition 3.5 (Leading-order term in the SO(3)-monopole cobordism formula (29) for link pairings)

(See F and Leness [8, Proposition 4.1].) Continue the notation and assumptions of Theorem 3.4. In addition, assume that there is K ∈ B(X ) with K 6= 0. Let m and n be non-negative integers satisfying the conditions (27). Deﬁne A := c(X ) − n − 2m − 1, and δ := c(X ) + 4χh(X ) − 3n − 1, and ` = 2χh(X ) − n. Then

(δ − 2m)! (30) a (χ (X ), c2(X ), 0, 0, m, `) = (−1)m+`2`−δ . A,0,` h 1 `!A!

68 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsV

The proof of Proposition 3.5 requires knowledge of the topology of an open neighborhood of

1 Ms × {x} ⊂ M¯t/S ,

where x ∈ Sym`(X ). While we generally need to consider arbitrary lower levels, ` ≥ 0, in the Uhlenbeck compactiﬁcation,

N ¯ G ` Mt ⊂ Mt(`) × Sym (X ), `=0

69 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsVI

the proof of Proposition 3.5 only requires us to consider points,

x ∈ Sym`(X ),

in the top stratum of Sym`(X ) (all points distinct). We can also show that

70 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsVII

Proposition 3.6 (Vanishing coeﬃcients in SO(3)-monopole cobordism formula for link pairings)

(See F and Leness [8, Proposition 5.1].) Continue the hypothesis and notation of Theorem 3.4. In addition, assume c(X ) ≥ 3 and

(31) n ≡ 1 (mod 2).

Then for p ≥ c(X ) − 3 and k ≥ 0 an integer such that p + 2k = c(X ) + 4χh(X ) − 3n − 1 − 2m,

2 ap,0,k (χh(X ), c1 (X ), 0, 0, m, 2χh − n) = 0.

71 / 132 Introduction and main results SO(3)-monopole equations Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism formula for link pairings SO(3)-monopole cobordism SO(3)-monopole cobordism and Donaldson invariants Superconformal simple type and Witten’s formula SO(3)-monopole cobordism and algebraic geometry Local and global gluing maps for SO(3) monopoles SO(3)-monopole cobordism and relations among SW invariants Bibliography SO(3)-monopole cobordism and SW invariant relationsVIII

We prove Theorem 1.2 (simple type =⇒ superconformal simple type) by applying the computations of the coeﬃcients in 2 Proposition 3.5 (formula for aA,0,`(χh(X ), c1 (X ), 0, 0, m, `)), 2 Proposition 3.6 (ap,0,k (χh(X ), c1 (X ), 0, 0, m, 2χh − n) = 0), to the vanishing sum formula (29), namely

2χh(X )−n X w,A+2k 2χh(X )−n+k 0 = aA+2k,0,2χh(X )−n−k SWX (h)QX (h) . k=0

72 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography

Superconformal simple type and Witten’s formula

73 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Blow-up formulaeI

We recall versions of the blow-up formulae for Donaldson and Seiberg-Witten invariants. These formulae are used to Verify invariance of Witten’s Formula (4) under blow-ups, Verify invariance of the Superconformal Simple Type property under blow-ups, Eliminate the need to consider certain diﬃcult cases, Enrich useful families of example manifolds for which Donaldson and Seiberg-Witten invariants are known.

74 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Blow-up formulaeII

Let Xe → X be the blow-up of X at one point, let e ∈ H2(Xe; Z) be the fundamental class of the exceptional curve, and let ∗ 2 e ∈ H (Xe; Z) be the Poincar´edual of e. Using the direct sum decomposition of the homology and 2 cohomology of Xe = X #CP , we can consider both the homology and cohomology of X as subspaces of those of Xe, • • H•(X ) ⊂ H•(X˜) and H (X ) ⊂ H (X˜). Denotew ˜ := w + e∗. The simplest blow-up formula for Donaldson invariants (see Kotschick [22] or Leness [30] for SO(3) invariants and Ozsv´ath[41] for SU(2) invariants) gives

(32) Dw (hδ−2mxm) = Dw˜ (hδ−2mexm). X Xe 75 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Blow-up formulaeIII

Versions of the blow-up formula for Seiberg-Witten invariants have been established by Fintushel and Stern [14], Nicolaescu [39, Theorem 4.6.7], and Frøyshov [18, Theorem 14.1.1] (in increasing generality). The following is a special case of their results.

76 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Blow-up formulaeIV

Theorem 4.1 (Blow-up formula for Seiberg-Witten invariants)

2 Let X be a standard four-manifold and let Xe = X #CP¯ be its blow-up. Then Xe has Seiberg-Witten simple type if and only if that is true for X . If X has Seiberg-Witten simple type, then

(33) B(Xe) = {K ± e∗ : K ∈ B(X )},

where e∗ ∈ H2(Xe; Z) is the Poincar´edual of the exceptional curve, and if K ∈ B(X ), then SW 0 (K ± e∗) = SW 0 (K). Xe X The signiﬁcance of Theorem 4.1 lies in its universality; more general versions, with more complicated statements, hold without the assumption of simple type.

77 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Refinements of SO(3)-monopole cobordism formulaI

We shall rewrite the SO(3)-monopole cobordism formula (25) for w δ−2m m 0 DX (h x )) as a sum over B (X ) ⊂ B(X ), a fundamental domain for the action of {±1}. To this end, we deﬁne (compare [12, Equation (4.4)])

2 2 bi,j,k (χh(X ), c1 (X ), K · Λ, Λ , m) c(X )+i 2 2 := (−1) ai,j,k (χh(X ), c1 (X ), −K · Λ, Λ , m) 2 2 + ai,j,k (χh(X ), c1 (X ), K · Λ, Λ , m),

where ai,j,k are the coeﬃcients appearing in the expression (26) for

2 fδ,m(χh(X ), c1 (X ), K, Λ)(h)

w δ−2m m in the SO(3)-monopole cobordism formula (25) for DX (h x ). 78 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Refinements of SO(3)-monopole cobordism formulaII

To simplify the orientation factor in (25), we deﬁne

˜ 2 2 (34) bi,j,k (χh(X ), c1 (X ), K · Λ, Λ , m) 1 (Λ2+Λ·K) 2 2 := (−1) 2 bi,j,k (χh(X ), c1 (X ), K · Λ, Λ , m).

Observe that

˜ 2 2 (35) bi,j,k (χh(X ), c1 (X ), −K · Λ, Λ , m) c(X )+i+Λ·K ˜ 2 2 = (−1) bi,j,k (χh(X ), c1 (X ), K · Λ, Λ , m).

We now rewrite the SO(3)-monopole cobordism formula (25) as a sum over B0(X ).

79 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Refinements of SO(3)-monopole cobordism formulaIII

Lemma 4.2 (SO(3)-monopole cobordism formula on fundamental domain)

(See F and Leness [10, Lemma 3.4].) Assume the hypotheses of Theorem 3.3 (the SO(3)-monopole cobordism formula). Denote the coeﬃcients in (35) more concisely by

˜ ˜ 2 2 bi,j,k (K · Λ) := bi,j,k (χh(X ), c1 (X ), K · Λ, Λ , m).

1 2 Then, for ε(w, K) = 2 (w + w · K) as in (22) and ν(K) as in (23),

w δ−2m m X X ε(w,K) 0 DX (h x ) = ν(K)(−1) SWX (K) K∈B0(X ) i+j+2k (36) =δ−2m i j k × b˜i,j,k (K · Λ)hK, hi hΛ, hi QX (h) .

80 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsI

To determine the coeﬃcients b˜i,j,k appearing in the w δ−2m m SO(3)-monopole cobordism formula (36) for DX (h x ), we compare w δ−2m m Witten’s formula (21) for DX (h x ), and w δ−2m m SO(3)-monopole cobordism formula (36) for DX (h x ), on manifolds where Witten’s Conjecture 1.1 is known to hold. In [13], Fintushel, Park and Stern constructed families of symplectic manifolds with one basic class (as rational blow-downs of elliptic surfaces). We use the Fintushel-Park-Stern manifolds to create a family of standard four-manifolds, Xq, for q = 2, 3,... , obeying the

81 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsII

following conditions (see F and Leness [12, Section 4.2] and [10, Section 4.3]):

1 Xq satisﬁes Witten’s Conjecture 1.1;

2 For q = 2, 3,... , one has χh(Xq) = q and c(Xq) = 3; 0 3 B (Xq) = {K} with K 6= 0; 2 4 For each q, there are classes f1, f2 ∈ H (Xq; Z) satisfying

2 (37a) f1 · f2 = 1, fi = 0, and fi · K = 0 for i = 1, 2, 2 {f1, f2, K} linearly independent subset of H (Xq; R),(37b)

Restriction of QXq to Ker f1 ∩ Ker f2 ∩ Ker K is non-zero.(37c)

82 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsIII

Let Xq(n) be the blow-up of Xq at n points,

2 2 (38) Xq(n) := Xq# CP # ··· #CP . | {z } n copies

Then Xq(n) is a standard four-manifold of Seiberg-Witten simple type and satisﬁes Witten’s Conjecture 1.1 (see F and Leness [12, Theorem 2.7] or [10, Theorem 2.3]), with

2 (39) χh(Xq(n)) = q, c1 (Xq(n)) = q − n − 3, and c(Xq(n)) = n + 3.

We will consider both the homology and cohomology of Xq as subspaces of those of Xq(n). 83 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsIV

∗ 2 Let eu ∈ H (Xq(n); Z) be the Poincar´edual of the u-th exceptional class, 1 ≤ u ≤ n. n Let πu :(Z/2Z) → Z/2Z be projection onto the u-th factor. n For ϕ ∈ (Z/2Z) , we deﬁne

n n X πu(ϕ) ∗ X ∗ (40) Kϕ := K + (−1) eu and K0 := K + eu . u=1 u=1

By the blow-up formula for Seiberg-Witten invariants (see Frøyshov [18, Theorem 14.1.1])

0 n (41) B (Xq(n)) = {Kϕ : ϕ ∈ (Z/2Z) },

84 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsV

n and, for all ϕ ∈ (Z/2Z) ,

(42) SW 0 (K ) = SW 0 (K). Xq(n) ϕ Xq

Because Xq(n) has Seiberg-Witten simple type, we have

2 2 n (43) Kϕ = c1 (Xq(n)) for all ϕ ∈ (Z/2Z) .

In addition, because K 6= 0, we see that

0 (44) 0 ∈/ B (Xq(n)).

0 For n ≥ 2, the set B (Xq(n)) is not a linearly independent subset 2 of H (Xq(n); R) but can be replaced by linearly independent ∗ ∗ ∗ subset, {K ± e1 , e2 ,..., en } to give the 85 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsVI

Lemma 4.3 (Donaldson invariants of Xq(n) via SO(3)-monopole cobordism)

(See F and Leness [10, Lemma 4.6].) For n, q ∈ Z with n ≥ 1 and q ≥ 2, let Xq(n) be 2 the manifold deﬁned in (38). For Λ, w ∈ H (Xq; Z) and δ, m ∈ N satisfying ˜ 2 Λ − w ≡ w2(Xq) (mod 2) and δ − 2m ≥ 0, deﬁne w˜, Λ ∈ H (Xq(n); Z) by

n n X ∗ ˜ X ∗ (45)w ˜ := w + wueu and Λ := Λ + λueu , u=1 u=1

where wu, λu ∈ Z and wu + λu ≡ 1 (mod 2) for u = 1,..., n. We assume that

n 2 X 2 (46a) Λ > δ − (n + 3) − 4q + λu, u=1 n 2 X 2 (46b) δ ≡ −w + wu − 3q (mod 4). u=1

86 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsVII

Lemma 4.3 (Donaldson invariants of Xq(n) via SO(3)-monopole cobordism) ˜ ˜ Denote x := Kϕ · Λ and, for i, j, k ∈ N satisfying i + j + 2k + 2m = δ, write

˜ ˜ 2 ˜2 bi,j,k (x) = bi,j,k (χh(Xq(n)), c1 (Xq(n)), x, Λ , m).

Then, for x0 = K0 · Λ˜ where K0 is deﬁned in (40),

87 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsVIII

Lemma 4.3 (Donaldson invariants of Xq(n) via SO(3)-monopole cobordism)

n ! X (δ − 2m)! Y pw˜ (i ,..., i ) he∗, hiiu Q (h)k k+n−m 2 n u Xq (n) 2 k!i1! ··· in! i1+···+in+2k u=2 =δ−2m ∗ i1 w1 ∗ i1 × hK + e1 , hi + (−1) hK − e1 , hi n ! i + ··· + in (47) X 1 ˜ j Y ∗ iu k = hΛ, hi heu , hi QXq (n)(h) i1,..., in i1+···+in+j+2k u=2 =δ−2m

i2+w2 in+wn ∗ i × ∇ · · · ∇ b˜ (x0)hK + e , hi 1 2λ2 2λn i,j,k 1

w i2+w2 in+wn ∗ i +(−1) 1 ∇ · · · ∇ b˜ (x0 + 2λ1)hK − e , hi 1 , 2λ2 2λn i,j,k 1

88 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsIX

Lemma 4.3 (Donaldson invariants of Xq(n) via SO(3)-monopole cobordism) are both equal to the following multiple of the Donaldson invariant,

(−1)ε(w ˜,ϕ0) Dw˜ (hδ−2mxm), SW 0 (K) Xq (n) Xq

where Λ˜ is as deﬁned in (45) and

( w˜ 0 if ∃u with 2 ≤ u ≤ n and wu + iu ≡ 1 (mod 2), (48) p (i2,..., in) = n−1 2 if wu + iu ≡ 0 (mod 2) ∀ u with 2 ≤ u ≤ n.

89 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsX

For a function f : Z → R and p, q ∈ Z, we deﬁne

q q (∇pf )(x) := f (x) + (−1) f (x + p), ∀ x ∈ Z.

The forthcoming Proposition 4.4 determines the coeﬃcients b˜i,j,k with i ≥ c(X ) − 3. However, the forthcoming condition (49a) in Proposition 4.4 prevents an immediate determination of the coeﬃcients b˜i,j,k with i < c(X ) − 3.

90 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Fintushel-Park-Stern family of example manifoldsXI

˜ Proposition 4.4 (Computation of coeﬃcients bi,j,k for i ≥ c(X ) − 3)

(See F and Leness [12, Proposition 4.8] or [10, Proposition 4.7].) Let n > 0 and q ≥ 2 be integers. If x, y are integers and i, j, k, m are non-negative integers satisfying, for A := i + j + 2k + 2m,

(49a) i ≥ n, (49b) y > A − 4q − 3 − n, (49c) A ≥ 2m, (49d) x ≡ y ≡ 0 (mod 2),

˜ 2 2 then the coeﬃcients bi,j,k (χh, c1 , Λ · K, Λ , m) deﬁned in (34) are given by  (A − 2m)! m−k−n ˜  2 if j = 0, bi,j,k (q, q − 3 − n, x, y, m) = k!i! 0 if j > 0.

91 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Blow-up trickI

The following lemma allows us to ignore the coeﬃcients b˜0,j,k in w δ−2m m the formula (36) for DX (h x ) for the purpose of proving Theorem 1.4 and Corollary 1.5.

92 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Reﬁnements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Diﬀerence equations and superconformal simple type Bibliography Blow-up trickII

Lemma 4.5 (Eliminating the coeﬃcients b˜0,j,k in the formula (36) for w δ−2m m DX (h x )) (See F and Leness [10, Lemma 3.5].) Continue the notation and hypotheses of Lemma 4.2. Then,

X X 2(i + 1) (50) Dw (hδ−2mx m) = (−1)ε(w,K)SW 0 (K) X X (δ − 2m + 1) K∈B0(X ) i+j+2k =δ−2m ˜ 2 2 × bi+1,j,k (χh(X ), c1 (X ) − 1, K · Λ, Λ , m) i j k × hK, hi hΛ, hi QX (h) .

93 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeI

Proposition 4.4 yields the coeﬃcients b˜i,j,k in the SO(3)-monopole w δ−2m m cobordism formula (36) for DX (h x ) with i ≥ c(X ) − 3 > 0

but not those coeﬃcients b˜i,j,k with

0 ≤ i < c(X ) − 3,

when c(X ) − 3 > 0.

Lemma 4.5 allows us to ignore the coeﬃcients b˜i,j,k with i = 0, that is, b˜0,j,k when proving Theorem 1.4 and Corollary 1.5.

94 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeII

However, again using the fact that the Fintushel-Park-Stern four-manifolds, Xq(n), satisfy Witten’s Conjecture 1.1, then w δ−2m m Witten’s formula (21) for DX (h x ), and w δ−2m m SO(3)-monopole cobordism formula (36) for DX (h x ), when applied to the manifolds Xq(n), will also show that, when 1 ≤ i < c(X ) − 3, the coeﬃcients b˜i,j,k not determined by Proposition 4.4 satisfy a homogeneous diﬀerence equation in the parameter K · Λ and thus can be written as polynomials in K · Λ.

95 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeIII

Proposition 4.6 (Diﬀerence equation for b˜i,j,k with 1 ≤ i < c(X ) − 3)

(See F and Leness [10, Proposition 4.9].) Let n > 1 and q ≥ 2 be integers. If x, y are integers and p, j, k, m are non-negative integers satisfying, for A := p + j + 2k + 2m,

(51a) 1 ≤ p ≤ n − 1, (51b) y > A − 4q − n − 3, (51c) y ≡ A − (n + 3) (mod 4), (51d) x − y ≡ 0 (mod 2),

and we abbreviate b˜p,j,k (x) = b˜p,j,k (q, q − n − 3, x, y, m), then

n−p 1 ˜ (52) ∇4 bp,j,k (x) = 0.

96 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeIV

We next record an algebraic consequence of superconformal simple type which will allow us to show that Witten’s Formula (4) holds even without determining the coeﬃcients b˜i,j,k with i < c(X ) − 3 w δ−2m m in the SO(3)-monopole cobordism formula (36) for DX (h x ).

97 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeV

Lemma 4.7 (An algebraic consequence of superconformal simple type)

(See F and Leness [10, Lemma 5.1].) Let X be a standard four-manifold of superconformal simple type. Assume 0 ∈/ B(X ). If w ∈ H2(X , Z) is characteristic and j, u ∈ N satisfy j + u < c(X ) − 3 and j + u ≡ c(X ) (mod 2), then

X ε(w,K) 0 j u (53) (−1) SWX (K)hK, h1i hK, h2i = 0, K∈B0(X )

for any h1, h2 ∈ H2(X ; R).

Proposition 4.6 and the diﬀerence equation for the coeﬃcients b˜i,j,k allow us to write the b˜i,j,k as polynomials in Λ · K.

98 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeVI

Corollary 4.8 (Coeﬃcients b˜i,j,k as polynomials in Λ · K)

(See F and Leness [10, Corollary 4.11].) Continue the assumptions of Proposition 4.6. In addition, assume

1 There is a class K0 ∈ B(X ) such that Λ · K0 = 0; 2 For all K ∈ B(X ), we have Λ · K ≡ 0 (mod 4).

Then for 1 ≤ i ≤ n − 1, the function b˜i,j,k is a polynomial of degree n − 1 − i in Λ · K and thus

n−1−i 2 X 2 u (54) b˜i,j,k (q, q −n −3, K ·Λ, Λ , m) = b˜u,i,j,k (q, q −n −3, Λ , m)hK, hΛi , u=0

where hΛ = PD[Λ] is the Poincar´edual of Λ and if u ≡ n + i (mod 2), then

2 (55) b˜u,i,j,k (q, q − n − 3, Λ , m) = 0.

99 / 132 Introduction and main results Blow-up formulae for Donaldson and Seiberg-Witten invariants Review of Donaldson and Seiberg-Witten invariants Refinements of SO(3)-monopole cobordism formula SO(3)-monopole cobordism Fintushel-Park-Stern family of example manifolds Superconformal simple type and Witten’s formula Blow-up trick Local and global gluing maps for SO(3) monopoles Difference equations and superconformal simple type Bibliography Difference equations and superconformal simple typeVII

We combine Corollary 4.8 with Lemma 4.7 to show that, for manifolds of superconformal simple type, the coeﬃcients b˜i,j,k with i ≤ c(X ) − 4 do not contribute to the SO(3)-monopole cobordism w δ−2m m expression (36) for the Donaldson invariant DX (h x ). This proves Theorem 1.4 (that Superconformal Simple Type =⇒ Witten’s Formula).

100 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography

Construction of local and global gluing maps and obstruction sections for SO(3) monopoles

101 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopolesI

When ` ≥ 1, the construction of links of Seiberg-Witten moduli subspaces, ` Ms × Sym (X ) ⊂ Mt, and the computation of intersection numbers for intersections of ∗,0 geometric representatives of cohomology classes on Mt with those links requires the construction of a (global) SO(3)-monopole gluing map (and obstruction section of an obstruction bundle, since gluing is always obstructed in the case of SO(3) monopoles). We summarize the steps in the construction of the local SO(3)-monopole gluing map and obstruction section and proofs of their properties and hence completing the veriﬁcation of

102 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopolesII Theorem 5.1 (Properties of local SO(3)-monopole gluing maps)

The local gluing map, constructed in [9], gives a continuous parametrization of a neighborhood of Ms × Σ in M¯t for each smooth stratum Σ ⊂ Sym`(X ).

These local gluing maps are the analogues for SO(3) monopoles of the local gluing maps for anti-self-dual SO(3) connections constructed by Taubes in [46, 47, 48], Donaldson [1], and Donaldson and Kronheimer in [3].

103 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopolesIII

Local splicing (or pregluing) map This map is a smooth embedding from the local gluing data parameter space — a finite-dimensional, open, Riemannian manifold — into the configuration space of gauge-equivalence classes of SO(3) pairs. The image of the map is given by gauge-equivalence classes of approximate SO(3) monopoles, [A, Φ], defined by a “cut-and-paste” construction. We splice anti-self-dual SU(2) connections from S4 onto background SO(3) monopoles on X (elements of Mt(`)) at points in the support of x ∈ Σ ⊂ Sym`(X )

104 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopolesIV to form gauge-equivalence classes of SO(3) pairs, [A, Φ], which are close to the stratum ¯ Mt(`) × Σ ⊂ Mt.

See F and Leness [5, 6, 9].

105 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopolesV

Local gluing map This is a smooth map from the gluing data parameter space deﬁned by a single stratum,

Σ ⊂ Sym`(X ),

into the conﬁguration space of SO(3) pairs. The image of the map is given by gauge-equivalence classes of extended SO(3)-monopoles,[A + a, Φ + φ], obtained by solving the extended SO(3)-monopole equations for the perturbations, (a, φ),

⊥ ΠA,φ,µS(A + a, Φ + φ) = 0,

106 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopoles VI

rather than the SO(3)-monopole equations directly,

S(A + a, Φ + φ) = 0,

since Coker DS(A, Φ) = Ran ΠA,Φ,µ is non-zero, where µ > 0 is a “small-eigenvalue” cut-off parameter. With respect to local coordinates and bundle trivializations, these equations comprise an elliptic, quasi-linear, partial integro-differential system. The gauge-equivalence classes of true SO(3) monopoles are given by the zero-locus of a local, smooth section of a finite-rank local Kuranishi obstruction bundle over the gluing data parameter space,

107 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical difficulties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopoles VII defined by L2-orthogonal projection onto finite-dimensional, “small-eigenvalue” vector spaces (see [5, 9]).

Smooth embedding property of the local gluing map One must compute the diﬀerential of the gluing map and prove that the diﬀerential is injective.

Surjectivity of the local gluing map Every extended SO(3) monopole close enough to the Uhlenbeck boundary of Mt must lie in the image of the local gluing map.

108 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Local gluing maps for SO(3) monopoles VIII

Continuity of the local gluing map and obstruction section The gluing map and obstruction section must extend continuously to the compactiﬁcation of the local gluing data space, which includes the Uhlenbeck compactiﬁcation of moduli spaces of anti-self-dual connections on S4.

109 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Global gluing maps for SO(3) monopolesI

Building a global gluing map and obstruction section from the local gluing maps and obstruction sections

Theorem 5.1 describes a neighborhood of Ms × Σ in M¯t for Σ ⊂ Sym`(X ) a smooth stratum while the proof of Theorem 3.3 (general SO(3)-monopole cobordism formula) requires a description ` of a neighborhood of the union of these strata, Ms × Sym (X ). In [6], we proved how the local gluing data parameter spaces, splicing maps, obstruction bundles, and obstruction sections given by Theorem 5.1 for different Σ ⊂ Sym`(X ) fit together and extend over the Uhlenbeck compactification, M¯t.

110 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Global gluing maps for SO(3) monopolesII

The splicing maps are deformed so that they obey a type of “cocycle condition” — to give global splicing maps and obstruction sections, solving the overlap problem identiﬁed by Kotschick and Morgan for gluing SO(3) anti-self-connections in [23]. Using this construction, we computed the expressions for the intersection number yielding the SO(3)-monopole cobordism formula (25) and completing the proof of Theorem 3.3.

111 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Analytical diﬃculties in gluing SO(3) monopolesI

Finally, we indicate some of the analytical diﬃculties in the construction of the gluing maps and obstruction sections that are particular to SO(3)-monopoles:

1 Small-eigenvalue obstructions to gluing. 1 1,∗ The Laplacian, dA,ΦdA,Φ, constructed from the diﬀerential, 1 dA,Φ = DS(A, Φ), of the SO(3)-monopole map S, at an almost SO(3) monopole, (A, Φ), has small eigenvalues tending to zero when A bubbles and S(A, Φ) tends to zero. This phenomenon occurs for SO(3) monopoles because the Dirac operators, when coupled with an anti-self-dual connections over S 4, always have non-trivial cokernels and

112 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Analytical diﬃculties in gluing SO(3) monopolesII

Seiberg-Witten monopoles need not be smooth points of their ambient moduli space of background SO(3) monopoles. 2 Bubbling curvature component in Bochner-Weitzenböck formulae. A key ingredient employed by Taubes in his solution to the anti-self-dual equation in [46, 47] is his use of the + +,∗ Bochner-Weitzenböckformula for the Laplacian, dA dA , + + constructed from the differential, dA , of the map, A 7→ FA , at an approximate anti-self-dual connection. While Taubes’ Bochner-Weitzenböckformula only involves the + small, self-dual curvature component, FA , our

113 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography Analytical diﬃculties in gluing SO(3) monopolesIII

1 1,∗ Bochner-Weitzenb¨ockformula for dA,ΦdA,Φ also involves the − large anti-self-dual curvature component, FA . 3 Seiberg-Witten moduli spaces of positive dimension and spectral ﬂow.

When dim Ms > 0, one cannot ﬁx a single, uniform positive 1 1,∗ upper bound for the small eigenvalues of dA,ΦdA,Φ, due to spectral ﬂow as the point [A, Φ] varies in an open ` neighborhood of Ms × Sym (X ) in the local gluing data parameter space. These are addressed in our preprints [9, 5, 7].

114 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants Construction of local gluing maps for SO(3) monopoles SO(3)-monopole cobordism Construction of global gluing maps for SO(3) monopoles Superconformal simple type and Witten’s formula Analytical diﬃculties in gluing SO(3) monopoles Local and global gluing maps for SO(3) monopoles Bibliography

Thank you for your attention!

115 / 132 Introduction and main results Review of Donaldson and Seiberg-Witten invariants SO(3)-monopole cobordism Superconformal simple type and Witten’s formula Local and global gluing maps for SO(3) monopoles Bibliography

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