Tree homology and a conjecture of Levine

Conant, James; Schneiderman, Rob; Teichner, Peter

doi:10.2140/gt.2012.16.555

Cited by 19 publications

(63 citation statements)

References 26 publications

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“…As explained in section 5 (Theorem 5.5), this result follows from the relationship between the Whitney tower intersection invariant τ and Milnor invariants [11,38], together with our proof in [12] of a combinatorial conjecture of J. Levine [29], which also implies that T 2k is a free abelian group of known rank [12, Thm.1.5].…”

Section: Ihxmentioning

confidence: 66%

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Whitney tower concordance of classical links

2012

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This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor's link invariants. 57M25, 57M27, 57Q60; 57N10

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Section: Ihxmentioning

confidence: 66%

“…As a first step towards our goal of describing a classification of this filtration and the associated W n , we recall (e.g. from [9,12,30,38]) a combinatorially defined group which is a natural target for the intersection invariant associated to the obstruction theory for Whitney towers.…”

Section: The Whitney Tower Filtration Of Classical Linksmentioning

confidence: 99%

Whitney tower concordance of classical links

2012

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“…Conversely, there is an epimorphism R n : T n → W n , called the realization map, such that R n (φ) is the class of a link bounding an order n framed Whitney tower T in D 4 withτ n (T ) = φ. For even n, they showed that T 2ℓ ∼ = Z M(m,n) where m is the number of link components, using their proof of the Levine conjecture [CST12a,CST12c]. (Recall that M(m, n) is the rank of D n ; see Remark 3.2.)…”

Section: Some Results From the Integral Framed Theorymentioning

confidence: 99%

“…In Theorem 3.1 (2), µ n denotes the total Milnor invariant of order n, and η n denotes the summation map which was formulated in [Lev01,Lev02] and used extensively in [CST12c,CST14,CST12a]. We will review their definitions in Section 3.1.…”

Section: Milnor Invariants and Rational Whitney Towersmentioning

confidence: 99%

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Rational Whitney tower filtration of links

Choon

2017

Math. Ann.

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Abstract. We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which are conjectured to be nontrivial, measure the potential difference between the Whitney tower theory in rational homology 4-balls and that in the 4-ball extensively developed by Conant, Schneiderman and Teichner. IntroductionTopology of dimension 4 is different from high dimensions because the Whitney move may fail. The essential problem is to find an embedded Whitney disk along which a pair of intersections of two sheets could be removed by a Whitney move. Once an immersed Whitney disk is obtained from fundamental group data, one may try to remove double points of the disk by finding a next stage of Whitney disks. Iterating this, we are led to the notion of a Whitney tower.Since work of Cochran, Orr and Teichner [COT03], concordance of knots and links, which is the "local case" of general disk embedding, has been extensively studied via frameworks formulated in terms of Whitney towers. In this paper, we will focus on asymmetric Whitney towers in dimension 4 bounded by links in S 3 , motivated from work of Conant, Schneiderman and Teichner [CST11,CST12c,CST14,CST12b,CST12a]. Whitney towers come in two flavors: framed and twisted. Whitney towers we consider have an order, which is a nonnegative integer measuring the number of iterated stages. Precise definitions can be found in Section 2.The main result of this paper is a complete classification of links in S 3 modulo Whitney towers in rational homology 4-balls. To state our result, we use the following notation. Fix m > 0, and let W n be the set of m-component links in S 3 bounding a twisted Whitney tower of order n in a rational homology 4-ball with boundary S 3 . We define the graded quotient W n of W n by the condition that L and L ′ in W n represent the same element in W n if and only if a band sum of L and −L ′ lies in W n+1 . In fact, in Section 4.3, we will show that it is an equivalence relation, and L ∈ W n+1 if and only if [L] = 0 in W n . So we may write W n = W n /W n+1 . Theorem A.(1) Band sum is a well-defined operation on the set W n , independent of the choice of bands, and W n is an abelian group under band sum. (2) W n is classified by the Milnor invariants of order n (= length n + 2). (3) W n is a free abelian group of rank mR(m, n + 1) − R(m, n + 2), where R(m, n) = We remark that R(m, n) is the rank of the degree n part of the free Lie algebra on m variables, due to Witt (e.g., see [MKS66, Section 5.6]), and mR(m, n + 1) − R(m, n + 2)2010 Mathematics Subject Classification. 57N13, 57N70, 57M25. Key words and phrases. Whitney towers, Concordance, Rational homology 4-balls, Links, Milnor invariants, higher order Arf invariants. RATIONAL WHITNEY TOWER FILTRATION OF LINKS2 is the number of linearly independent Milnor invar...

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Lagrangian traces for the Johnson filtration of the handlebody group

Faes

2023

Topology and its Applications

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Tree homology and a conjecture of Levine

Cited by 19 publications

References 26 publications

Whitney tower concordance of classical links

Whitney tower concordance of classical links

Rational Whitney tower filtration of links

Lagrangian traces for the Johnson filtration of the handlebody group

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