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13
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.
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 4 (4 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.
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+
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.
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 of ..."
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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.
IN S p+q+r+1
"... Let f: S p × S q × S r smooth embedding. → Sp+q+r+1, 2 ≤ p ≤ q ≤ r, be a In this paper we show that the closure of one of the two components of Sp+q+r+1 − f(S p × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, provided that p + q ̸ = r or p + q = r w ..."
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Let f: S p × S q × S r smooth embedding. → Sp+q+r+1, 2 ≤ p ≤ q ≤ r, be a In this paper we show that the closure of one of the two components of Sp+q+r+1 − f(S p × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, provided that p + q ̸ = r or p + q = r with r even. We also show that when p + q = r with r odd, there exist infinitely many embeddings which do not satisfy the above property. We also define standard embeddings of S p × S q × S r into S p+q+r+1 and, using the above result, we prove that if C1 has the homology of S p ×S q, then f is standard, provided that q<r. 1. Introduction. In [A], Alexander has shown that a piecewise linearly embedded torus in the
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
SOME REMARKS ON SYMMETRIC PRODUCTS OF CURVES
"... Abstract. Symmetric products of curves are important spaces for both geometers and topologists, and increasingly useful objects for physicists. We summarize below some of their basic homotopy theoretic properties and derive a handful of known and less wellknown results about them. We combine in a s ..."
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Abstract. Symmetric products of curves are important spaces for both geometers and topologists, and increasingly useful objects for physicists. We summarize below some of their basic homotopy theoretic properties and derive a handful of known and less wellknown results about them. We combine in a slightly leisurely way both geometry and topology to describe some useful properties of symmetric products of algebraic curves. We record a straightforward derivation of Clifford’s theorem from a calculation of MacDonald, point to an equally simple characterization of hyperelliptic curves and discuss the embeddability (both in the continuous and holomorphic categories) of the unique spherical generator in dimension two in the homology of these spaces. A homotopy retract statement about the AbelJacobi map is also proven. 1. Cohomology Structure and Clifford’s Theorem Given a complex algebraic curve C and n ≥ 1, the nth symmetric product of C is the quotient C (n) = C n /Σn, where Σn is the symmetric group acting on C n by permuting coordinates 1. Elements in C (n) are referred to as effective divisors on C. A point D ∈ C (n) is said to have degree n and we can write it as a formal linear combination ∑ nixi where xi ̸ = xj ∈ C for i ̸ = j, and ni are positive integers with ∑ ni = n. 1.1. MacDonald and Clifford’s theorems. It it assumed wellknown that C (n) is a complex (smooth) algebraic variety for all positive n. The nth AbelJacobi map is an algebraic map µn: C (n) −−−→J(C) where J(C) is the “Jacobian ” of C (a complex torus of dimension the genus of C). It is additive in the sense that the following commutes
INERTIA GROUPS OF SMOOTH EMBEDDINGS
, 2005
"... Abstract. We work in the smooth category. Let N be a closed connected nmanifold and assume that m> n + 2. Denote by Emb m (N) the set of embeddings N → S m up to isotopy. The group Emb m (S n) acts on Emb m (N) by embedded connected sum of a manifold and a sphere. If Emb 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 Emb m (N) the set of embeddings N → S m up to isotopy. The group Emb m (S n) acts on Emb m (N) by embedded connected sum of a manifold and a sphere. If Emb m (S n) is nonzero (which often happens for 2m < 3n + 4) then no results on this action and no complete description of Emb m (N) were known. Our main results are examples of the triviality and the effectiveness of this action for embeddings of 4manifolds into S 7. The proofs are based on the first author’s modification of surgery theory and on the invention of a new embeding invariant. As a corollary we obtain the following complete description of Emb 7 (N). If n = 4, H1(N; Z) = 0 and the signature σ(N) of N is not divisible by a square of an integer s> 1, then there is a 1–1 correspondence Emb 7 (N) = {x ∈ H2(N; Z)  x mod 2 = P.D.w2(N), x 2 = σ(N)}. 1. Introduction and