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31
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 115 (7 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 24 (5 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
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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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.
D Ruberman, Topologically slice knots with nontrivial Alexander polynomial
 Adv. Math. 231 (2012) 913–939 34 ARUNIMA RAY
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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 16 (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)
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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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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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...
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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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
Filtering smooth concordance classes of topologically slice knots, preprint
, 2012
"... Abstract. We propose and analyze a structure with which to organize the difference between a knot in S3 bounding a topologically embedded 2disk in B4 and it bounding a smoothly embedded disk. The nsolvable filtration of the topological knot concordance group, due to CochranOrrTeichner, may be co ..."
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Abstract. We propose and analyze a structure with which to organize the difference between a knot in S3 bounding a topologically embedded 2disk in B4 and it bounding a smoothly embedded disk. The nsolvable filtration of the topological knot concordance group, due to CochranOrrTeichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the nsolvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, {Bn}, that is simultaneously a refinement of the nsolvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn/Bn+1 has infinite rank. But our primary interest is in the induced filtration, {Tn}, on the subgroup, T, of knots that are topologically slice. We prove that T /T0 is large, detected by gaugetheoretic invariants and the τ, s, invariants; while the nontriviliality of T0/T1 can be detected by certain dinvariants. All of these concordance obstructions vanish for knots in T1. Nonetheless, going beyond this, our main result is that T1/T2 has positive rank. Moreover under a “weak homotopyribbon ” condition, we show that each Tn/Tn+1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity. 1.