1-loop graphs and configuration space integral for embedding spaces

Sakai, Keiichi; Watanabe, Tadayuki

doi:10.1017/s0305004111000429

Cited by 7 publications

(11 citation statements)

References 18 publications

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“…As we mentioned earlier all our graph-complexes look very similar to the graph-complexes that appear in the Bott-Taubes type integration construction for spaces of long embeddings [9,32,33,42]. The latter construction produces a map from a certain graph-complex to the de Rham complex of differential forms on Emb c (R m , R n ).…”

Section: 3supporting

confidence: 60%

“…Similarly we obtain two different complexes computing Q ⊗ π * (Emb c (R m , R n )), see Section 2 and 5. It is quite interesting that all the obtained complexes computing the rational homology and homotopy of Emb c (R m , R n ) look very similar to the graph-complexes arising in the Bott-Taubes integration for the space of long knots and their higher dimensional analogues [8,9,32,33,42]. In the paper we also determine how the rational homotopy type of Emb c (R m , R n ) is related to that of Emb c (R m , R n ), see Section 4.…”

mentioning

confidence: 74%

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Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

Arone¹,

Turchin²

2015

Annales De l'Institut Fourier

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We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on different ways how the calculations can be done. In particular we describe three different graph-complexes computing these rational homotopy groups. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

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Section: 3supporting

confidence: 60%

mentioning

confidence: 74%

Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

Arone¹,

Turchin²

2015

Annales De l'Institut Fourier

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“…In this and the forthcoming papers [16] we will develop the methods to study K n,j originated in perturbative Chern-Simons theory. This method was used by various authors [1,3,10] to give some integral expressions of the finite type invariants for knots in R 3 .…”

Section: Introductionmentioning

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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“…The map I yields many nonzero cohomology classes even in the non-stable dimensions and in some dimensions not necessarily satisfying the condition in Theorem 4.2; for example, H 3 DR (K 5,2 ) 0. See [26,33].…”

Section: ]) the Map I Is A Cochain Map Ifmentioning

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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1-loop graphs and configuration space integral for embedding spaces

Cited by 7 publications

References 18 publications

Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant

Lin–Wang type formula for the Haefliger invariant

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