Milnor invariants and twisted Whitney towers

Conant, James; Schneiderman, Rob; Teichner, Peter

doi:10.1112/jtopol/jtt025

Cited by 23 publications

(85 citation statements)

References 46 publications

(156 reference statements)

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“…See Theorem 3.1. This generalizes an earlier result in [CST14].The elimination of the higher order Arf invariants in the rational theory generalizes an earlier result too. Indeed, the figure eight knot, which has nontrivial Arf invariant, is known to bound a slice disk in a rational homology 4-ball [Cha07], and this tells us that the classical Arf invariant is not preserved under rational concordance.…”

supporting

confidence: 89%

“…see [CST14, Section 4.3]). Also, the Milnor invariant of order n gives rise to a homomorphism µ n : W n → D n [CST14].…”

Section: Some Results Of the Integral Twisted Theorymentioning

confidence: 99%

“…In this section we will review definitions of twisted and framed asymmetric Whitney towers in 4-manifolds, and discuss uni-trivalent trees which arise naturally in the study of iterated intersections of surfaces, particularly for Whitney towers (e.g., see [Coc90,CT04a,CT04b,Sch06,CST07,CST12c,CST14]). Readers who are familiar with them may skip to Section 3, after reading this paragraph.…”

Section: Whitney Towers and Associated Treesmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Especially Igusa and Orr proved the k-slice conjecture, which asserts that a link L has vanishing Milnor invariants of length ≤ 2k if and only if L bounds disjoint surfaces in D 4 such that each loop on these surfaces can be pushed off to a loop lying in the kth lower central subgroup of the fundamental group of the complement of the surfaces [IO01]. A significantly strengthened version of the Igusa-Orr theorem was given in [CST14, Theorem 18] by Conant, Schneiderman and Teichner.As a consequence of our main result, we present a geometric characterization of the vanishing of the Milnor invariants in terms of Whitney towers. …”

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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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supporting

confidence: 89%

“…see [CST14, Section 4.3]). Also, the Milnor invariant of order n gives rise to a homomorphism µ n : W n → D n [CST14].…”

Section: Some Results Of the Integral Twisted Theorymentioning

confidence: 99%