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21
Higher order intersection numbers of 2spheres in 4manifolds
 ALGEBRAIC & GEOMETRIC TOPOLOGY
, 2000
"... This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection nu ..."
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This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection number µ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if µ(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of π1X modulo S3symmetry (rather then just one copy modulo S2symmetry). It generalizes to the nonsimply connected setting the KervaireMilnor invariant defined in [2] and [12] which corresponds to the Arfinvariant of knots in 3space. We also give necessary and sufficient conditions for homotoping three maps f1, f2, f3: S2 → X to a position in which they have disjoint images. The obstruction λ(f1, f2, f3) generalizes Wall’s intersection number λ(f1, f2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in 3space, our new invariant corresponds to the Milnor invariant µ(1, 2, 3), generalizing the Matsumoto triple to the non simplyconnected setting. Finally, we explain some simple algebraic properties of these new cubic forms on π2(X) in Theorem 3. These are straightforward generalizations of the properties of quadratic forms as defined by Wall [14, §5]. A particularly attractive formula is λ(f, f, f) = ∑ τ(f) σ σ∈S3 which generalizes the well known fact that Wall’s invariants satisfy λ(f, f) = µ(f) + µ(f) = ∑ µ(f) σ for an immersion f with trivial normal bundle. σ∈S2 1.
Primary decomposition and the fractal nature of knot concordance
, 2009
"... Abstract. For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3, such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higherorder Alexander ..."
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Abstract. For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3, such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higherorder Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the (n)solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higherorder analogues of the p(t)primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no CochranOrrTeichner knot is concordant to any CochranHarveyLeidy knot. 1.
Simple Whitney towers, halfgropes and the Arf invariant of a knot
 Pacific Journal of Mathematics
, 2005
"... Abstract. A geometric characterization of the Arf invariant of a knot in the 3–sphere is given in terms of two kinds of 4–dimensional bordisms, halfgropes and Whitney towers. These types of bordisms have associated complexities class and order which filter the condition of bordism by an embedded an ..."
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Cited by 11 (10 self)
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Abstract. A geometric characterization of the Arf invariant of a knot in the 3–sphere is given in terms of two kinds of 4–dimensional bordisms, halfgropes and Whitney towers. These types of bordisms have associated complexities class and order which filter the condition of bordism by an embedded annulus, i.e. knot concordance, and it is shown constructively that the Arf invariant is exactly the obstruction to cobording pairs of knots by halfgropes and Whitney towers of arbitrarily high class and order. This illustrates geometrically how, in the setting of knot concordance, the Vassiliev (isotopy) invariants “collapse ” to the Arf invariant. 1.
2TORSION IN THE nSOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP
, 907
"... Abstract. CochranOrrTeichner introduced in [11] a natural filtration of the smooth knot concordance group C · · · ⊂ Fn+1 ⊂ Fn.5 ⊂ Fn ⊂ · · · ⊂ F1 ⊂ F0.5 ⊂ F0 ⊂ C, called the (n)solvable filtration. We show that each associated graded abelian group {Gn = Fn/Fn.5  n ∈ N}, n ≥ 2 contains inf ..."
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Abstract. CochranOrrTeichner introduced in [11] a natural filtration of the smooth knot concordance group C · · · ⊂ Fn+1 ⊂ Fn.5 ⊂ Fn ⊂ · · · ⊂ F1 ⊂ F0.5 ⊂ F0 ⊂ C, called the (n)solvable filtration. We show that each associated graded abelian group {Gn = Fn/Fn.5  n ∈ N}, n ≥ 2 contains infinite linearly independent sets of elements of order 2 (this was known previously for n = 0, 1). Each of the representative knots is negative amphichiral, with vanishing sinvariant, τinvariant, δinvariants and CassonGordon invariants. Moreover each is slice in a rational homology 4ball. In fact we show that there are many distinct such classes in Gn, one for each “distinct ” ntuple P = (p1(t),..., pn(t)) of knot polynomials. Such a sequence of polynomials records the orders of certain submodules of a sequence of higherorder Alexander modules of the knot. 1.
Spherical Lagrangians via ball packings and symplectic cutting, arXiv:1211.5952
"... In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S2 or RP2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff’s connectedness of ball packings in other set ..."
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In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S2 or RP2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floertheoretically essential Lagrangian tori in the del Pezzo surfaces. 1
JHOLOMORPHIC CURVES IN ALMOST COMPLEX SURFACES DO NOT ALWAYS MINIMIZE THE GENUS
"... Abstract. The adjunction formula computes the genus of an almost complex curve F embedded in an almost complex surface M in terms of the homology class of F. If M is Kähler (or at least symplectic) and the selfintersection of F is nonnegative then the genus of any other surface embedded in M and h ..."
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Abstract. The adjunction formula computes the genus of an almost complex curve F embedded in an almost complex surface M in terms of the homology class of F. If M is Kähler (or at least symplectic) and the selfintersection of F is nonnegative then the genus of any other surface embedded in M and homologous to F is not less then the genus of F (the proof of this statement (which is a generalization of the Thom conjecture for CP 2) was recently given by the SeibergWitten theory). This paper shows that the extra assumptions on M are essential for the genusminimizing properties of embedded almost complex curves. Let M be a connected smooth 4manifold with the tangent bundle τM equipped with a fiberwiselinear map J: τM → τM respecting the fibers and such that J 2 = −1. In this case M is called an almost complex surface. TheRlinear map J makes τM into a 2dimensional complex bundle and induces an orientation on M. The canonical class K ∈ H2 (M; Z) is the Euler class of the exterior square over C of τM multiplied by (−1). An embedded surface F ⊂ M is called a Jholomorphic curve if its tangent bundle τF is invariant under J. AJholomorphic curve gets an orientation from J. If Fis Jholomorphic then the normal bundle νF can be chosen to be invariant under J, and the direct sum formula for the characteristic classes of bundles produces the adjunction formula for the genus g(F)ofF: F.F + K.F g(F)=1+, 2 where F.F denotes the selfintersection of F and K.F denotes the result of evaluation of K on F. Let E ⊂ M be an orientable surface homologous to F.Thegenusg(E)ofEis not determined by its homology class. However, in the case when M is symplectic with the symplectic form ω compatible to J so that ω(x, Jx) ≥ 0 for any x ∈ τM and F.F is nonnegative, the adjunction formula turns into the adjunction inequality F.F + K.F g(E) ≥ g(F)=1+
Homotopy classification of ribbon tubes and welded string links, preprint
, 2014
"... Abstract. Ribbon 2knotted objects are locally flat embeddings of surfaces in 4–space which bound immersed 3manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider rib ..."
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Abstract. Ribbon 2knotted objects are locally flat embeddings of surfaces in 4–space which bound immersed 3manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon tubes, which are knotted annuli bounding ribbon 3balls. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to linkhomotopy, that is when allowing each tube component to cross itself. At the combinatorial level, this provides a classification of welded string links up to selfvirtualization. This generalizes a result of Habegger and Lin on usual string links, and the abovementioned action on the reduced free group can be refined to a general “virtual extension ” of Milnor invariants. We also give a classification of ribbon toruslinks up to linkhomotopy. Finally, connections between usual, virtual and welded knotted objects are investigated. Dedicated to Eléonore, Lise, Helena and Siloé.
A CLASSIFICATION OF SMOOTH EMBEDDINGS OF 4MANIFOLDS IN 7SPACE, I
, 2008
"... Abstract. We work in the smooth category. Let N be a closed connected nmanifold and assume that m> n + 2. Denote by E m (N) the set of embeddings N → R m up to isotopy. The group E m (S n) acts on E m (N) by embedded connected summation of a manifold and a sphere. If E m (S n) is nonzero (which ..."
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Abstract. We work in the smooth category. Let N be a closed connected nmanifold and assume that m> n + 2. Denote by E m (N) the set of embeddings N → R m up to isotopy. The group E m (S n) acts on E m (N) by embedded connected summation of a manifold and a sphere. If E m (S n) is nonzero (which often happens for 2m < 3n + 4) then until recently no results on this action and no complete description of E m (N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R 7 for certain 4manifolds N. The proofs use new approach based on the Kreck modified surgery theory and the construction of a new invariant. Corollary. (a) There is a unique embedding f: CP 2 → R 7 up to isoposition (i.e. for each two embeddings f, f ′ : CP 2 → R 7 there is a diffeomorphism h: R 7 → R 7 such that f ′ = h ◦ f). (b) For each embedding f: CP 2 → R 7 and each nontrivial embedding g: S 4 → R 7 the embedding f#g is isotopic to f. 1.