High-codimensional knots spun about manifolds

Roseman, Dennis; Takase, Masamichi

doi:10.2140/agt.2007.7.359

Cited by 3 publications

(3 citation statements)

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“…Remark 4.10. In [13] a generator of π 0 (Emb (S 4k−1 , S 6k )) was defined by the deform-spinning ψ : (S 2k−1 ) 2 → K 2k+2,1 along the torus. But it is not difficult to see that such a spinning also gives an element of K 6k,4k−1 which is isotopic to our S ψ given by the graphing map [4].…”

Section: 2mentioning

confidence: 99%

Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant

Sakai

2010

J. Knot Theory Ramifications

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Abstract. Let K n,j be the space of long j-knots in R n . In this paper we introduce a graph complex D * and a linear map I : D * → Ω * DR (K n,j ) via configuration space integral, and prove that (1) when both n > j ≥ 3 are odd, I is a cochain map if restricted to graphs with at most one loop component, (2) when n − j ≥ 2 is even, I is a cochain map if restricted to tree graphs, and (3) when n − j ≥ 3 is odd, I added a correction term produces a (2n − 3j − 3)-cocycle of K n,j which gives a new formulation of the Haefliger invariant when n = 6k, j = 4k − 1 for some k.

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Section: 2mentioning

confidence: 99%

Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant

Sakai

2010

J. Knot Theory Ramifications

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“…In [25] Theorem 4.3 is proved by evaluating H over a generator of π 0 (K 6k,4k−1 ) given by the spinning construction [5,24]. The computation in §3 gives an alternative proof.…”

Section: The Haefliger Invariantmentioning

confidence: 99%

Lin–Wang type formula for the Haefliger invariant

Sakai

2015

Homology, Homotopy and Applications

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In this paper we study the Haefliger invariant for long embeddings R 4k−1 ֒→ R 6k in terms of the self-intersections of their projections to R 6k−1 , under the condition that the projection is a generic long immersion R 4k−1 R 6k−1 . We define the notion of "crossing changes" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in R 4k−1 . This formula is a higher-dimensional analogue to that of X.-S. Lin and Z. Wang for the order 2 invariant for classical knots. As a consequence, we show that the Haefliger invariant is of order two in a similar sense to Birman and Lin. We also give an alternative proof for the result of M. Murai and K. Ohba concerning "unknotting numbers" of embeddings R 3 ֒→ R 6 . Our formula enables us to define an invariant for generic long immersions R 4k−1 R 6k−1 which are liftable to embeddings R 4k−1 ֒→ R 6k . This invariant corresponds to V. Arnold's plane curve invariant in Lin-Wang theory, but in general our invariant does not coincide with order 1 invariant of T. Ekholm.Date: October 13, 2018. Key words and phrases. the space of embeddings, the Haefliger invariant, configuration space integral, finite type invariant, generic immersion.

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“…The generator has also been described (Theorem 3.13) as an iterated graphing construction applied to r, the resolution of an immersion of R in Euclidean space, corresponding to the chord-diagram (see Cattaneo, Cotta-Ramusino and Longini [15]). More recently, another spinning construction involving r has recently been developed by Roseman and Takase [62].…”

Section: Question 512mentioning

confidence: 99%

A family of embedding spaces

Budney¹

2008

Geometry and Topology Monographs

116

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Let Emb(S^j,S^n) denote the space of C^infty-smooth embeddings of the j-sphere in the n-sphere. This paper considers homotopy-theoretic properties of the family of spaces Emb(S^j,S^n) for n >= j > 0. There is a homotopy-equivalence of Emb(S^j,S^n) with SO_{n+1} times_{SO_{n-j}} K_{n,j} where K_{n,j} is the space of embeddings of R^j in R^n which are standard outside of a ball. The main results of this paper are that K_{n,j} is (2n-3j-4)-connected, the computation of pi_{2n-3j-3} (K_{n,j}) together with a geometric interpretation of the generators. A graphing construction Omega K_{n-1,j-1} --> K_{n,j} is shown to induce an epimorphism on homotopy groups up to dimension 2n-2j-5. This gives a new proof of Haefliger's theorem that pi_0 (Emb(S^j,S^n)) is a group for n-j>2. The proof given is analogous to the proof that the braid group has inverses. Relationship between the graphing construction and actions of operads of cubes on embedding spaces are developed. The paper ends with a brief survey of what is known about the spaces K_{n,j}, focusing on issues related to iterated loop-space structures.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

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High-codimensional knots spun about manifolds

Cited by 3 publications

References 21 publications

Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant

Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant

Lin–Wang type formula for the Haefliger invariant

A family of embedding spaces

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