Jacobi identities in low-dimensional topology

Conant, James; Schneiderman, Rob; Teichner, Peter

doi:10.1112/s0010437x06002697

Cited by 21 publications

(76 citation statements)

References 22 publications

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“…The reader familiar with the orientation and sign conventions of [21] can check by inserting signs and orientations in the figures that the following construction actually replaces an "I" tree with the difference "H − X" as in the usual IHX relation of finite type theory. Note that this differs from the closely related 4-dimensional IHX construction in [6] which creates the trees I − H + X for a Whitney tower on 2-spheres in 4-space by modifying the boundaries of Whitney disks.…”

Section: Proof Of Corollary 3 and The Whitney Move Ihx Constructionmentioning

confidence: 97%

“…the roots in t(g i ) will always correspond to i-labelled univalent vertices of t(W) when passing between gropes g i and a Whitney tower W on order zero surfaces A i . (These isomorphisms also preserve the signed trees associated to gropes and Whitney towers as in [6] and [21] Assume now that n ≥ 2. The proof will be completed by the following construction which shows how to decrease the order of g W while increasing the class of g W in a manner that preserves the tree t(g W ).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…[15]) can check that our definition of t(g c ) is essentially what you would get after applying Krushkal's grope-splitting procedure. In fact, a formal sum of vertex-oriented trees can be associated to an oriented grope ( [6] [7] [8]); although we will not work with orientations in this paper, the notation here has been chosen to be compatible with the just cited works.…”

Section: 2mentioning

confidence: 99%

“…In light of these notational conventions, it turns out to be convenient to associate to each grope a disjoint union of unitrivalent trees (as in [6], [7] and [8]) rather than the customary (single) multivalent tree (e.g. in [15]).…”

Section: Gropesmentioning

confidence: 99%

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Whitney towers and gropes in 4–manifolds

Schneiderman

2005

Trans. Amer. Math. Soc.

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113

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Abstract. Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated 'towers' of Whitney disks. The 'flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to n-solvability, k-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.

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Section: Proof Of Corollary 3 and The Whitney Move Ihx Constructionmentioning

confidence: 97%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Gropesmentioning

confidence: 99%

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Whitney towers and gropes in 4–manifolds

Schneiderman

2005

Trans. Amer. Math. Soc.

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113

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“…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%

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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Embedding calculus and grope cobordism of knots

Kosanović

2024

Advances in Mathematics

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Jacobi identities in low-dimensional topology

Cited by 21 publications

References 22 publications

Whitney towers and gropes in 4–manifolds

Whitney towers and gropes in 4–manifolds

Rational Whitney tower filtration of links

Embedding calculus and grope cobordism of knots

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