Whitney towers and the Kontsevich integral

Schneiderman, Rob; Teichner, Peter

doi:10.2140/gtm.2004.7.101

Cited by 20 publications

(101 citation statements)

References 36 publications

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“…Intersection invariants for (twisted) Whitney towers. By [12,49], an order n (twisted) Whitney tower W built on properly immersed disks in the 4-ball has an intersection invariant τ n (W) (resp. τ n (W)) which is defined by associating unitrivalent trees to the unpaired higher-order intersection points (and twisted Whitney disks) in W (e.g.…”

Section: 2mentioning

confidence: 99%

“…In the Whitney tower obstruction theory of [49], the order n intersection invariant τ n (W) ∈ T n assigned to each order n (framed) Whitney tower W is defined by summing the trees associated to unpaired intersections in W (see Figure 2 for an example). The tree orientations are induced by Whitney disk orientations via a convention that corresponds to the AS relations (Section 2.5), and the IHX relations can be realized geometrically by controlled maneuvers on Whitney towers as described in [10,47].…”

Section: 2mentioning

confidence: 99%

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Milnor invariants and twisted Whitney towers

Conant

Schneiderman

Teichner

2013

Journal of Topology

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This paper describes the relationship between the first non‐vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2‐complex in the 4‐ball, built from immersed disks bounded by the given link in the 3‐sphere together with finitely many ‘layers’ of Whitney disks. The intersection invariant is a higher‐order generalization of the intersection number between two immersed disks in the 4‐ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks. This interpretation of Milnor invariants as higher‐order intersection invariants plays a key role in our classifications [J. Conant, R. Schneiderman and P. Teichner, ‘Higher‐order intersections in low‐dimensional topology’, Proc. Natl Acad. Sci. USA 108 (2011) 8131–8138; J. Conant, R. Schneiderman and P. Teichner, ‘Whitney tower concordance of classical links’, Geom. Topol. 16 (2012) 1419–1479] of both the framed and twisted Whitney tower filtrations on link concordance. Here, we show how to realize the higher‐order Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing length at most 2k Milnor invariants.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Milnor invariants and twisted Whitney towers

Conant

Schneiderman

Teichner

2013

Journal of Topology

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“…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: 98%

“…Such enhancements are used in the obstruction theory of [20], [21], [22], and [23]. Notation in this paper has been chosen to be consistent with these papers where they overlap.…”

Section: Whitney Towersmentioning

confidence: 99%

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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“…Note that the intersection and self‐intersection numbers

λ_{M}, μ_{M}

are considered as lying at order zero, whereas

τ_{M}

is of order one. There are invariants of all orders (with large indeterminacies in their target groups), defined in an inductive way, as described by Schneiderman and the fourth author [, Definition 9]. The idea is as follows: if some algebraic count of intersections vanishes, pair these intersections up by Whitney discs, and count how these new Whitney discs intersect the previous surfaces.…”

Section: Introductionmentioning

confidence: 99%

Stable classification of 4-manifolds with 3-manifold fundamental groups

Kasprowski

Land

Powell

et al. 2017

Journal of Topology

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We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class.Comment: 55 pages, 2 figure

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Whitney towers and the Kontsevich integral

Cited by 20 publications

References 36 publications

Milnor invariants and twisted Whitney towers

Milnor invariants and twisted Whitney towers

Whitney towers and gropes in 4–manifolds

Stable classification of 4-manifolds with 3-manifold fundamental groups

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