## Gauge theory for embedded surfaces (1993)

Venue: | I, Topology |

Citations: | 67 - 6 self |

### BibTeX

@ARTICLE{Kronheimer93gaugetheory,

author = {P. B. Kronheimer and T. S. Mrowka},

title = {Gauge theory for embedded surfaces},

journal = {I, Topology},

year = {1993},

volume = {32},

pages = {773--826}

}

### Years of Citing Articles

### OpenURL

### Abstract

(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

### Citations

1362 |
Principles of algebraic geometry
- Griffiths, Harris
- 1978
(Show Context)
Citation Context ... X is the underlying 4-manifold of a complex surface with c1 = 0, and if Σ is a smooth holomorphic curve, then the inequality of the theorem is sharp (this is a special case of the adjunction formula =-=[GH]-=-). The simply-connected complex surfaces with c1 = 0 are the K3 surfaces, and it is one of the main results of [D3] that the hypothesis of Theorem 1.1 on the non-vanishing of the invariants is satisfi... |

474 |
The geometry of fourmanifolds
- Donaldson, Kronheimer
- 1990
(Show Context)
Citation Context ... ¯ L) 2 . Although this calculation is global, it has an interpretation locally on N. Let Yε ⊂ N be the 3-manifold circle bundle over Σ given by r = ε, and consider the Chern-Simons integral (see eg. =-=[DK]-=-) τε(A α )= 1 8π 2 � Yε tr(dA∧A+ 2 3 A∧A∧A). The integral depends only on the homotopy class of the trivialization of the bundle on Yε with respect to which the connection matrix A is computed. But on... |

465 |
The index of elliptic operators
- Atiyah, Segal
- 1986
(Show Context)
Citation Context ...⊓⊔ Remark. There are other ways to prove the formula. For example, having used excision to reduce to the case of an orbifold which is globally a quotient, one can apply the general G-index theorem of =-=[AS]-=-. Another possibility is to use excision to reduce to the case of a Kähler orbifold, where the cohomology groups can be identified with sheaf cohomology groups associated with a bundle with parabolic ... |

444 |
L.Tu: Differential Forms in Algebraic Topology
- Bott
- 1982
(Show Context)
Citation Context ...egrates on Σ to give Σ·Σ, the degree of the normal bundle. Since the second cohomology of the neighbourhood is 1-dimensional, it follows that this closed 2-form represents the Poincaré dual of Σ (see =-=[BT]-=- for this construction of the Thom class). In de Rham cohomology we therefore have i i [dbα]= 2π 2π [d¯b]+α[Σ] =c1( ¯ L)+α[Σ]. Denoting the form on the left-hand side by ω, we now calculate the left h... |

389 |
Singular Points of Complex Hypersurfaces
- Milnor
- 1968
(Show Context)
Citation Context ...section number of the class. The result applies to those 4-manifolds for which the polynomial invariants [D3] are defined and not all zero. As a corollary, we shall answer a question raised by Milnor =-=[Mi]-=- concerning the unknotting number of algebraic knots. This first paper contains statements of the results and most of the technical material; the sequel provides the more geometrical parts of the proo... |

212 |
The topology of four-dimensional manifolds
- Freedman
- 1982
(Show Context)
Citation Context ...elf-intersection number is negative, for the inequality is then satisfied automatically except in the one case that is explicitly excluded. Note finally that it is a consequence of Freedman’s theorem =-=[F]-=- that the smoothness assumption cannot be weakened to allow locally flat, topologically embedded surfaces; indeed, if X is spin for example, then every primitive class in H2(X) is represented by a loc... |

134 | An application of gauge theory to four-dimensional topology - Donaldson - 1983 |

110 |
Polynomial invariants for smooth four manifolds, Topology 29
- Donaldson
- 1990
(Show Context)
Citation Context ...el [KrM] is to establish a lower bound for the genus of the surface, in terms of the self-intersection number of the class. The result applies to those 4-manifolds for which the polynomial invariants =-=[D3]-=- are defined and not all zero. As a corollary, we shall answer a question raised by Milnor [Mi] concerning the unknotting number of algebraic knots. This first paper contains statements of the results... |

80 |
Moduli of vector bundles on curves with parabolic structures
- Seshadri
- 1977
(Show Context)
Citation Context ...f stable bundles with parabolic structure in the case that X is a Kähler surface and Σ is a holomorphic curve; see [Kr] for a statement of a conjecture modeled on the 2-dimensional result of Seshadri =-=[Se]-=-, also proved in [Bi2]. A weakened form of this result (for the orbifold metric) is proved in [KrM], and while this is adequate for the applications, the conjecture as stated in [Kr] remains attractiv... |

66 |
Instantons and four-manifolds
- Freed, Uhlenbeck
- 1984
(Show Context)
Citation Context ...of Vp ,thesectionΦwhichcuts out (Mα k,l )∗ is Fredholm; (iii) the index of Φ is 8k +4l−3(b + − b 1 +1)−(2g − 2). The third proposition of this section is a generic metrics theorem, modeled on that of =-=[FU]-=-. As usual, we note that the anti-self-duality equations can be defined using metrics which are, say, of class C r for some r; such metrics are parametrized by an open subset of a Banach space, and th... |

36 |
Some cobordism invariants for links
- Tristram
- 1969
(Show Context)
Citation Context ...] give a lower bound for the genus of a non-primitive surface in terms of the self-intersection number Σ·Σ and the invariants of X. In the case of the 4-manifold CP 2 , similar bounds are obtained in =-=[Tr]-=-. More recent results [MMR] have established further obstructions to embedding spheres and tori in 4-manifolds for which the polynomial invariants are non-zero; these can be seen as extensions of the ... |

30 | Principles of algebraic geometry - Griths, Harris - 1978 |

26 |
Classification of singular Sobolev connections by their holonomy
- Sibner, Sibner
- 1992
(Show Context)
Citation Context ...ent of Σ was gauge equivalent to a solution in our space; this would make the definition of our moduli spaces look less arbitrary. The authors have not proved such a theorem; a result on the lines of =-=[SS]-=- is what is wanted. A second point concerns the choice of metric on X. For large parts of this paper we shall not use a smooth metric, but choose instead a metric with a cone-like singularity along Σ ... |

23 | Torelli theorems for Kähler K3 surfaces - Looijenga, Peters - 1981 |

22 |
Connections, cohomology and the intersection forms of four manifolds
- Donaldson
- 1986
(Show Context)
Citation Context ...tions. The antiself-dual subspace for the metric g can be represented as the graph of a unique linear map µ : Λ − → Λ + defined by a bundle map whose pointwise operator norm is everywhere less than 1 =-=[D2]-=-. We shall suppose that there is a uniform upper bound |µ| which is strictly less than 1. The map µ encodes the conformal structure of g (see [D2]), so we can dispense with g and think only of µ as gi... |

18 | Fibrés parabolique stables et connexions singulières plates - Biquard |

16 | Pseudofree orbifolds - Fintushel, Stern - 1985 |

13 | Di erential Forms in Algebraic Topology - Bott, Tu - 1982 |

12 | Knot theory of complex plane curves
- Rudolph
- 2005
(Show Context)
Citation Context ...r such knots, the unknotting number and the genus of the Milnor fibre are equal, which confirms the conjecture made in [Mi]. A useful survey of conjectures related to (1.3) is given in [BW]; see also =-=[Ru]-=-. Apart from the challenging problem of extending (1.2) to other complex surfaces, there are some modest extensions one could envisage making to Theorem 1.1. For example, the condition on the fundamen... |

11 |
Le problème de J. Milnor sur le nombre gordien des noeuds algébriques, Enseign
- Boileau, Weber
- 1984
(Show Context)
Citation Context .... It is also known that, for the knots which arise from singularities in algebraic curves, the genus of the Milnor fibre is an upper bound for the un4 P.B. Kronheimer and T.S. Mrowka knotting number =-=[BW]-=- (for example, in the case of Kp,q, a little experimentation shows that passing the curve through itself 1 2 (p−1)(q −1) times is sufficient). It therefore follows from Corollary 1.3 that, for such kn... |

10 | New results in the theory of four-dimensional manifolds - Rohlin - 1952 |

9 |
Two-dimensional submanifolds of four dimension manifolds
- Rohlin
- 1971
(Show Context)
Citation Context ...S]. In the case that the given homology class is not primitive (so is an integer multiple of another class), there are obstructions also to embedding surfaces of higher genus: the results of [HS] and =-=[R2]-=- give a lower bound for the genus of a non-primitive surface in terms of the self-intersection number Σ·Σ and the invariants of X. In the case of the 4-manifold CP 2 , similar bounds are obtained in [... |

8 |
Path connected Yang-Mills moduli spaces
- TAUBES
- 1984
(Show Context)
Citation Context ...the formal dimensions of the moduli spaces is four times this quantity, so the result follows. ⊓⊔ (ii) Proof of Proposition 8.1. The formalities of our proof are modeled on an argument used by Taubes =-=[T1]-=- to study minimizing sequences for the Yang-Mills functional. The technical side of the argument is rather simpler here, since we are dealing with anti-self-dual solutions. We present the argument in ... |

7 |
Prolongment d’un Fibre Holomorphe Hermitien a Courbure L p sur une Courbe Ouverte
- Biquard
- 1992
(Show Context)
Citation Context ... parabolic structure in the case that X is a Kähler surface and Σ is a holomorphic curve; see [Kr] for a statement of a conjecture modeled on the 2-dimensional result of Seshadri [Se], also proved in =-=[Bi2]-=-. A weakened form of this result (for the orbifold metric) is proved in [KrM], and while this is adequate for the applications, the conjecture as stated in [Kr] remains attractive. Acknowledgements. T... |

7 | On 2–spheres - Kervaire, Milnor - 1961 |

6 |
On manifolds homeomorphic to
- Kotschick
(Show Context)
Citation Context ...ly be relaxed, and (perhaps with very little change in the proof) one should expect to prove the same inequality for 4-manifolds such as the Barlow surface, using the invariant developed by Kotschick =-=[Ko]-=-. (ii) Twisted connections. A natural strategy for the class of problems discussed above is to study gauge theory (connections and the anti-self-duality equations) on the complement of the surface. A ... |

6 |
Embedded surfaces in 4-manifolds
- Kronheimer
- 1990
(Show Context)
Citation Context ...nding moduli spaces will not be the same as those which result from a cylindrical end, and one of the first problems is to develop the usual tools of gauge theory in this setting. In an earlier paper =-=[Kr]-=-, the first author wrote down some conjectures about the features of the moduli spaces for the incomplete metric and outlined a scheme whereby these might be applied to prove results such as Theorem 1... |

5 |
Exact, fractionally charged self-dual solution
- Forgács, Horváth, et al.
- 1981
(Show Context)
Citation Context ... print, as far as the authors are aware, is an anti-self-dual connection belonging, in our notation, to the moduli space M 1/4 (S 4 ,S 2 ), for the standard round pair; this solution was described in =-=[FHP]-=-, and was later generalized by the same authors to give solutions with arbitrary α. We shall write down a solution on T 2 × D 2 ,whereT 2 is a torus and D 2 is a disk, both with their standard metrics... |

5 |
On embedding surfaces in four-manifolds, in: Algebraic topology (Proc
- Hsiang, Szczarba
- 1970
(Show Context)
Citation Context ... [D1,Ku,FS]. In the case that the given homology class is not primitive (so is an integer multiple of another class), there are obstructions also to embedding surfaces of higher genus: the results of =-=[HS]-=- and [R2] give a lower bound for the genus of a non-primitive surface in terms of the self-intersection number Σ·Σ and the invariants of X. In the case of the 4-manifold CP 2 , similar bounds are obta... |

5 | On Diffeomorphisms of a K3 Surface - Matumoto - 1985 |

3 | Representing homology classes of - Kuga - 1984 |

3 |
Monopoles and spectral curves for arbitrary Lie groups
- Murray
- 1983
(Show Context)
Citation Context ...= κ+λ λ ′ = −λ which is effected by twisting the bundle by −1, as in Lemma 2.12; so proving the necessity of either inequality implies them both. The conjecture is analogous to a result due to Murray =-=[Mu]-=- on the admissible values for the magnetic charges of non-abelian magnetic monopoles. Proposition 8.1 will be proved below. We have stated it above with the smooth metric in mind, but it is valid also... |

3 | On manifolds homeomorphic to CP - Kotschick - 1989 |

2 |
L moduli spaces on manifolds with cylindrical ends
- Taubes
- 1990
(Show Context)
Citation Context ... exp2πi , 0 α with α ∈ (0, 1 2 ). Since the circle’s class is central in π1(Yε) and its nth power is a product of commutators, the representation must be reducible on Yε and α must be of the form a/n =-=[T2]-=-. We can regard A now as one of our model connections corresponding to some pair (k,l). The formula for the cohomology class of the form ω in the proof of the previous proposition gives 〈ω,Σ〉 = −l+αΣ·... |

1 |
D.M.: Locally flat 2-spheres in simply-connected 4manifolds
- Lee, Wilczynski
- 1990
(Show Context)
Citation Context ...ssumption cannot be weakened to allow locally flat, topologically embedded surfaces; indeed, if X is spin for example, then every primitive class in H2(X) is represented by a locally flat sphere (see =-=[LW]-=-). The inequality in the theorem says that the sum of the Euler numbers of Σ and its normal bundle is not greater than zero. Alternatively, we can think of the decomposition of TX into a sum of orient... |

1 |
Gauge theory and involutions, D.Phil thesis
- Wang
- 1990
(Show Context)
Citation Context ... were introduced by Biquard in [Bi1,Bi2], and our constructions in section 3 are motivated in part by this example. The other model which guides our constructions is provided by the thesis of S. Wang =-=[W]-=-. Rather than use Lp spaces, one might try to use weighted Sobolev spaces based on L2 . Unfortunately, it seems that any straightforward attempt to introduce such spaces runs foul of the necessary mul... |