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15
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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 ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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
Representations of Orbifold Groups and Parabolic Bundles
, 1990
"... Contents 1 Introduction 1 2 Orbifolds 4 3 Parabolic Bundles 9 4 Push Forward Construction 15 5 Main Theorem 22 6 Applications 32 6.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 The filtration on C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 Th ..."
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Cited by 14 (4 self)
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Contents 1 Introduction 1 2 Orbifolds 4 3 Parabolic Bundles 9 4 Push Forward Construction 15 5 Main Theorem 22 6 Applications 32 6.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 The filtration on C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 The gauge groups G C and P . . . . . . . . . . . . . . . . . . . . . . . 34 6.4 The equivariant cohomology of C ss . . . . . . . . . . . . . . . . . . . 35 6.5 The cohomology of S in the case C ss = C s . . . . . . . . . . . . . . . 37 6.6 Results for genus 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.7 Relationship between S and R(\Sigma): . . . . . . . . . . . . . . . . . . . 42 6.8 Explicit computations . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1 Introductio
Smooth group actions on definite 4manifolds and moduli spaces
 Duke Math. J
, 1995
"... In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this ..."
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Cited by 12 (2 self)
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In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by an equivariant connected sum along an orbit of isolated points. Given a smooth oriented 4manifold X which is diffeomorphic to a connected sum of algebraic surfaces, we can ask: (i) which (finite) groups can act smoothly on X preserving the orientation, and (ii) how closely does a smooth action on X resemble some equivariant connected sum of algebraic actions on the algebraic surface factors of X? For the purposes of this paper we will restrict our attention to the simplest case, namely X p2(C) #... # p2(C), a connected sum of n copies of the complex projective plane (arranged so that X is simply connected). Furthermore, ASSUMPTION. All actions will be assumed to induce the identity on H,(X, Z). In previous works [17], [18], [19], we considered problem (i) and a variant of problem (ii) when X p2(C). It turned out that the only finite groups which could act as above on p2(C) were the subgroups of PGL3(C) ([18] and [23] independently). For problem (ii) there are 2 interesting notions weaker than smooth equivalence. If (X, r) is a smooth action, then the isotropy group rx { 9 rclOx x}, x X, acts linearly on the tangent space TxX and we can ask the following. Question (iii) a. Given an action (X, n), is there an equivariant connected sum of actions on p2(C) with the same fixed point data and tangential isotropy representations? Question (iii) b. Given an action (X, re), is there an equivariant connected sum of actions on p2(C) which is nhomotopy equivalent or nequivariantly homeomorphic to (X, n)? Partial results were obtained on these questions in [17] and [10]: if n acts smoothly on p2(C), inducing the identity on homology, and the action has an
Unitary Representations of Brieskorn Spheres
 Duke Math J
, 1993
"... In this article, we commence an investigation of the SU(N) representation space of Seifert fibered homology spheres \Sigma(a 1 ; : : : ; an ): Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of \Sigma(a 1 ..."
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Cited by 11 (7 self)
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In this article, we commence an investigation of the SU(N) representation space of Seifert fibered homology spheres \Sigma(a 1 ; : : : ; an ): Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of \Sigma(a 1 ; : : : ; an ) is homeomorphic to a component of SU(N) representations of an associated genus zero Fuchsian group. The latter representation spaces can be studied using the general correspondence between representations of Fuchsian groups and the moduli of parabolic bundles given by Mehta and Seshadri. For example, the inductive procedure of AtiyahBottNitsure determines the cohomology of this moduli space and it follows that the odd dimensional cohomology groups of any component of irreducible SU(N) representations of \Sigma(a 1 ; : : : ; an ) vanish. In particular, any irreducible component of the SU(3) representation space of a Brieskorn spheres \Sigma(p; q; r) is either a point or a two...
The sliceribbon conjecture for 3stranded pretzel knots
, 2007
"... Abstract. We determine the (smooth) concordance order of the 3stranded pretzel knots P(p, q, r) with p, q, r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the sliceribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtain ..."
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Cited by 9 (2 self)
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Abstract. We determine the (smooth) concordance order of the 3stranded pretzel knots P(p, q, r) with p, q, r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the sliceribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by FintushelStern and CassonGordon.
Is there a topological Bogomolov–Miyaoka–Yau inequality
"... Let S be a smooth, complex, projective, minimal surface of general type. The Bogomolov–Miyaoka–Yau inequality states that c1(S) 2 ≤ 3c2(S) [Bog78, Rei78, Miy77, Yau77]. In this note I want to address the following question: Is there a topological analog of the Bogomolov–Miyaoka–Yau inequality? The 1 ..."
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Cited by 8 (0 self)
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Let S be a smooth, complex, projective, minimal surface of general type. The Bogomolov–Miyaoka–Yau inequality states that c1(S) 2 ≤ 3c2(S) [Bog78, Rei78, Miy77, Yau77]. In this note I want to address the following question: Is there a topological analog of the Bogomolov–Miyaoka–Yau inequality? The 11/8conjecture [Mat82, Fur01] can be viewed as such, but in Section 1 I write down another possible variant. Section 2 explores its relationship with the Montgomery–Yang problem on differentiable circle actions on S 5 and Section 3 examines its connection with the H–cobordism group of 3–manifolds. Related examples and questions on algebraic surfaces are discussed in Section 4. The last section studies the remarkable series of hypersurfaces (x a1 1 x2 + x a2 2 x3 + · · · + x an−1 n−1 xn + x an n x1 = 0)
A TWO COMPONENT LINK WITH ALEXANDER POLYNOMIAL ONE IS CONCORDANT TO THE HOPF
, 2004
"... Let L be a two component link in S 3, an embedding of two disjoint circles which is topologically locally flat, that is, which extends to an embedding of two solid tori. The link has Alexander polynomial one, if, and only if, the first homology of the universal abelian cover of the complement of the ..."
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Cited by 4 (0 self)
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Let L be a two component link in S 3, an embedding of two disjoint circles which is topologically locally flat, that is, which extends to an embedding of two solid tori. The link has Alexander polynomial one, if, and only if, the first homology of the universal abelian cover of the complement of the link vanishes. If the link has Alexander polynomial one, then the linking number of the two components is one. Two links are concordant if there is an topologically locally flat embedding which restricts to the given links when i = 0, 1. (S 1 ∐ S 1) × I → S 3 × I (S 1 ∐ S 1) × {i} → S 3 × {i} Theorem. A two component link with Alexander polynomial one is concordant to the Hopf link. Michael Freedman showed that a knot with Alexander polynomial one is concordant to the unknot [1], [2, p. 210]. Jonathan Hillman had a program for proving the above theorem (see [4, Section 7.6], which corrected the account in [3]), but asked whether a final obstruction must vanish. In this note we complete Hillman’s program and show that the surgery problem can be chosen so that the final obstruction vanishes. Topological surgery in dimension four is used; note that the fundamental group of the complement of the Hopf link is Z 2, which is a good group in the sense of [2]. The above theorem is not true in the
Permutations, isotropy and smooth cyclic group actions on definite 4–manifolds
, 2004
"... We use the equivariant Yang–Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4–manifolds. ..."
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Cited by 4 (1 self)
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We use the equivariant Yang–Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4–manifolds.
A characterisation of the n〈1〉⊕〈3〉 form and applications to rational homology spheres
"... Abstract. We conjecture two generalisations of Elkies ’ theorem on unimodular quadratic forms to nonunimodular forms. We give some evidence for these conjectures including a result for determinant 3. These conjectures, when combined with results of Frøyshov and of Ozsváth and Szabó, would give a si ..."
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Cited by 4 (3 self)
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Abstract. We conjecture two generalisations of Elkies ’ theorem on unimodular quadratic forms to nonunimodular forms. We give some evidence for these conjectures including a result for determinant 3. These conjectures, when combined with results of Frøyshov and of Ozsváth and Szabó, would give a simple test of whether a rational homology 3sphere may bound a negativedefinite fourmanifold. We verify some predictions using Donaldson’s theorem. Based on this we compute the fourball genus of some Montesinos knots. 1.
A topological method to compute spectral flow
 Kyungpook Math. J
, 1998
"... This paper describes a topological method to compute the spectral flow of a family of twisted Dirac operators, it includes two detailed examples. Briefly, a formula of Atiyah, Patodi and Singer expresses the spectral flow in terms of ChernSimons invariants and rho invariants. The first step is to c ..."
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Cited by 1 (0 self)
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This paper describes a topological method to compute the spectral flow of a family of twisted Dirac operators, it includes two detailed examples. Briefly, a formula of Atiyah, Patodi and Singer expresses the spectral flow in terms of ChernSimons invariants and rho invariants. The first step is to construct a flat cobordism to a new bigger 3manifold. The advantage of the new connection is that it is in the path component of a reducible connection. The second step is to calculate the effect of these operations on the invariants. The final step is an application of the Gsignature theorem to compute the invariants. The spectral flow of a family of operators is a generalization of the signature of a selfadjoint finite dimensional operator. If At is a family of operators with a real, discrete, spectrum, then the spectral flow of At is the number of eigenvalues that move from negative to positive minus the number which move from positive to negative. Recalling that the number of positive eigenvalues minus the number of negative eigenvalues, we see that SF(At) = Sign A1 − Sign A0