Whitney Trick Research Articles

The relative Whitney trick and its applications

We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a 4-manifold with boundary.Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological 4-manifold. We further prove that any link in a homology sphere is order k Whitney tower concordant to a link in \(S^3\) for all k. Finally, we explore the minimum Gordian distance from a link in \(S^3\) to a homotopically trivial link. Extending this notion to links in homology spheres, we use the relative Whitney trick to make explicit computations for 3-component links and establish bounds in general.

The Legendrian Whitney trick

In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.

Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints

Using the authors’ 2014 “constraint method,” we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that the “ r -fold Whitney trick” of Mabillard and Wagner (2014/2015) implies that the Topological Tverberg Conjecture for r -fold intersections fails dramatically for all r that are not prime powers.

A triple-point Whitney trick

We use a triple-point version of the Whitney trick to show that ornaments of three orientable [Formula: see text]-manifolds in [Formula: see text], [Formula: see text], are classified by the [Formula: see text]-invariant. A very similar (but not identical) construction was found independently by I. Mabillard and U. Wagner, who also made it work in a much more general situation and obtained impressive applications. The present note is, by contrast, focused on a minimal working case of the construction.

Morse moves in flow categories

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent index and handle cancellation which removes two critical points connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a Morse function or of a Floer functional. The Cohen-Jones-Segal construction associates a stable homotopy type to a framed flow category whose cohomology is designed to recover the corresponding Morse or Floer cohomology. We obtain analogues of the Whitney trick and of handle cancellation for framed flow categories: in this new setting these are moves that can be performed to simplify a framed flow category without changing the associated stable homotopy type. These moves often enable one to compute by hand the stable homotopy type associated to a framed flow category. We apply this in the setting of the Lipshitz-Sarkar stable homotopy type (corresponding to Khovanov cohomology) and the stable homotopy type of a matched diagram due to the authors (corresponding to sln Khovanov-Rozansky cohomology).

Mechanical normal forms of knots and flat knots

A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the original knot diagram has a small number of crossings, thus outperforming the famous M\"obius energy due to Jun O'Hara and studied by Michael Freedman et al. Various properties of the device are described (under certain conditions it does the Reidemeister and Markov moves, it beautifully performs the Whitney trick by uniformizing its own local curvature). If the device is constrained between two parallel planes (e.g. glass panes), it yields a real life model of a flat knot (class of knot diagrams equivalent under Reidemeister $\Omega_2$ and $\Omega_3$ moves) also leading to uniquely determined "flat normal forms" (for a small number of crossing points of the given flat knot diagram). The paper concludes with two mathematical theorems, one reducing the knot recognition problem to the flat knot recognition problem, the other (due to S.V.Matveev) giving an easily computable complete system of invariants for the flat unknot knot equivalence problem.

Four-dimensional Manifolds

Four-dimensional Manifolds

Embeddings in generalized manifolds

We prove that a ( 2 m − n + 1 2m-n+1 )-connected map f : M m → X n f\colon M^m\to X^n from a compact PL m m -manifold M M to a generalized n n -manifold X X with the disjoint disks property, 3 m ≤ 2 n − 2 3m\le 2n-2 , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact M M and proper maps f f that are properly ( 2 m − n + 1 2m-n+1 )-connected. The techniques developed lead to a general position result for arbitrary maps f : M → X f\colon M\to X , 3 m ≤ 2 n − 2 3m\le 2n-2 , and a Whitney trick for separating P L PL submanifolds of X X that have intersection number 0, analogous to the well-known results when X X is a manifold.

The Whitney Trick

The Whitney Trick is a procedure by which submanifolds of a given manifold can, under certain conditions, be isotoped off each other by using an embedded 2-disc. It works well when the ambient manifold is of dimension five or greater. Here, a particularly simple example of its breakdown in dimension four is exhibited.

An explicit family of exotic Casson handles

This paper contains a proof that the Casson handle that contains only one, positive, self-intersection on each level, CH+ , is exotic in the sense that the attaching circle of this Casson handle is not smoothly slice in its interior. The proof is an easy consequence of L. Rudolph's result (Bull. Amer. Math. Soc. (N.S.) 29 (1993), 51-59) that no iterated positive untwisted doubles of the positive trefoil knot is smoothly slice. An explicit infinite family of Casson handles is constructed by using the non-product h-cobordism from Z. Bilaca (A handle decomposition of an exotic R4, J. Differential Geom. (to appear)), CH,,, n > 0, such that CHO is the above-described CH+ and each CH,+, is obtained by the reimbedding algorithm (2. Bilaca, A reimbedding algorithm for Casson handles, Trans. Amer. Math. Soc. 345 (1994), 435-510) in the first six levels of CH, . An argument that no two of those exotic Casson handles are diffeomorphic is outlined, and it mimics the one from S. DeMichelis and M. Freedman (J. Differential Geom. 17 (1982), 357-453). The purpose of this paper is to present explicit examples of exotic Casson handles. For a description of Casson handles the reader is referred to [C], [F], or [K2]. The notion of 'exoticness' is defined below. A way to describe any handle, and so a Casson handle in particular, is to think of it as a pair consisting of a manifold and a selected piece of its boundary, "the attaching area". One of the biggest break-throughs in four-dimensional topology was M. Freedman's proof [F] that every Casson handle is homeomorphic to the standard open 2-handle, (D2 x R2, Si x R2). Andrew Casson constructed these manifold pairs in an attempt to mimic the proof of the h-cobordism theorem for higher-dimensional manifolds in the setting of four-dimensional manifolds [C]. The piece of the higher-dimensional proof that does not apply to dimension four is the Whitney trick (see [RS, ?6]). Casson handles were constructed in places where embedded 2-handles, or "Whitney discs", are needed in order to perform the Whitney trick [C]. Therefore, a consequence of Freedman's result is the topological hcobordism theorem for four-dimensional manifolds [F]; that is, an h-cobordism between two simply-connected closed four-dimensional manifolds, say MO and M, , is necessarily homeomorphic to the product cobordism, MO x [0, 1]. If all Casson handles were diffeomorphic to the standard open 2-handle, the Received by the editors July 28, 1993. 1991 Mathematics Subject Classification. Primary 57M99, 57N 13, 57R65.

A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion

Forn≥3, the ordinary Whitney trick shows that a simplicial complexKn having van Kampen obstruction classo(kn)=0 embeds inR2n. We give a one-dimensional version of the Whitney trick, by means of which any graphK1 satisfyingo(K1)=0 can be, step by step, embedded inR2. We then deduce some other planarity criteria, including Kuratowski’s, from this result. As a byproduct we also obtain a fascinating description of the mod 2 homology of the deleted product of a graph.

Link homotopy with one codimension two component

Link maps with one codimension two component are studied and an invariant of link maps modulo link homotopy is constructed using ideas from knot theory and immersion theory. This invariant is used to give examples of nontrivial link homotopy classes and to show that there are infinitely many distinct link homotopy classes in many dimensions. A link map with the codimension two component embedded is shown to be nullhomotopic. These ideas are applied to the special case of 2 2 -spheres in S 4 {S^4} to give simple examples of the failure of the Whitney trick in dimension 4 4 .

Finding Stable-Boundaries for Open Five-Dimensional Manifolds

Introduction. In 1965 W. Browder, J. Levine and G. R. Livesay characterized those smooth open n-manifolds, n > 5, which are interiors of compact manifolds with simply-connected boundary. (See [1].) Shortly afterward L. Siebenmann identified an invariant for a tame end of any smooth open n-manifold W, n ? 5. He showed that for n > 5, W is the interior of a compact manifold if and only if W has finitely many ends, each tame with invariant zero. (See [13].) The failure in dimension four of the well-known Whitney Trick prevents one from extending Siebenmann's arguments in order to obtain this result for arbitrary open 5-manifolds. Although the failure of Whitney's Trick prevents us from solving many problems involving 4-manifolds, it is widely known that useful information can often be obtained after stabilization; i.e. upon adding copies of S2 X 52 to the 4-manifolds by connected-sum. A noteworthy example is the 5-dimensional Stable S-cobordism Theorem. (See Quinn [12] or Lawson [9].) For other examples consult Wall [17] or Cappell and Shaneson [2]. In this paper we prove that if W is any smooth open 5-manifold having finitely many ends, each tame with invariant zero, then after adding a certain number of copies of 52 X 52 X R by connected-sum along suitable smooth arcs, the resulting open 5-manifold is the interior of a smooth compact manifold. Using different techniques M. Freedman and F. Quinn [5] have recently shown that if W is an open topological 5-manifold such that H*(W; Z) is finitely-generated and each end of W is 1-connected then W is the interior of a compact manifold whose boundary is an F4-manifold. (An F4-manifold is a homology 4-manifold with isolated 1-LC non-manifold points.) At present there is no corresponding nonsimply-connected theorem.

Milnor's [formula omitted]-invariants, Massey products and Whitney's trick in 4 dimensions

We present the failure of Whitney's lemma in dimension 4 from the homotopical and topological viewpoints. Those are detected by Massey products. The invariants for the examples represented by framed links are computed in terms of Milnor's μ -invariants.

Whitney's Trick for Three 2-Dimensional Homology Classes of 4-Manifolds

Whitney's Trick for Three 2-Dimensional Homology Classes of 4-Manifolds