## Coformality and the rational homotopy groups of spaces of long knots (2007)

Citations: | 5 - 4 self |

### BibTeX

@TECHREPORT{Arone07coformalityand,

author = {Greg Arone and Pascal Lambrechts and Victor Turchin and Ismar Voli Ć},

title = {Coformality and the rational homotopy groups of spaces of long knots},

institution = {},

year = {2007}

}

### OpenURL

### Abstract

Abstract. We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in R d for d ≥ 4 collapses at the E 2 page. The main ingredients in the proof are Sinha’s cosimplicial model for the space of long knots and a coformality result for the little balls operad. 1.

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Citation Context ...mark. It is well-known that for a group-like cosimplicial object, totalization is homotopy equivalent to “homotopy totalization”, which is defined to be the homotopy limit of the cosimplicial diagram =-=[4]-=-. In particular, totalization of a cosimplicial simplicial Abelian group always has the “right” homotopy type, and the same is true for totalization of cosimplicial chain complexes. There are natural ... |

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Citation Context ...to a multiplicative operad and hence by [14, Theorem 3.3] their totalizations admit an action of the little 2-balls operad. Since the totalizations are connected, by the loop recognition principle of =-=[13]-=-, they are in particular Hspaces. Therefore their rational homotopy type is determined by their rational homology because rational H-spaces are products of Eilenberg-MacLane spaces. □ We can now final... |

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Citation Context ...j < i ≤ n} of Hd−1(K(n);k) where γij is the homology class representing the fundamental class of the (d − 1)-dimensional sphere (coming from “point i turning around point j”) in R d , as was shown in =-=[7]-=-. On the other hand, Cohen and Gitler have shown in [6, Theorem 2.3] that there is an explicit isomorphism of DGLs (5) χ(n) ∼ = L(Bij : 1 ≤ j < i ≤ n)/I where the Bij are of degree d − 2 and I is the ... |

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Citation Context ...In particular K(n) is homotopy equivalent to the configuration space of n points in R d . An important additional feature is that it is a multiplicative operad in the following sense. Definition 4.2 (=-=[10, 15]-=-). Let (C, ⊗,1) be a monoidal category where 1 is the unit object for the monoidal operation ⊗. A multiplicative operad O(•) = {O(n)}n≥0 in C is a non-Σ operad (that is, an operad without the action o... |

57 | AA Voronov, Homotopy G-algebras and moduli space operad
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Citation Context ...In particular K(n) is homotopy equivalent to the configuration space of n points in R d . An important additional feature is that it is a multiplicative operad in the following sense. Definition 4.2 (=-=[10, 15]-=-). Let (C, ⊗,1) be a monoidal category where 1 is the unit object for the monoidal operation ⊗. A multiplicative operad O(•) = {O(n)}n≥0 in C is a non-Σ operad (that is, an operad without the action o... |

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Citation Context ...xes is that it corresponds, under the Dold-Kan functor from chain complexes to simplicial Abelian groups, to the usual totalization of cosimplicial simplicial spaces. This is essentially Lemma 2.2 of =-=[2]-=-. Remark. It is well-known that for a group-like cosimplicial object, totalization is homotopy equivalent to “homotopy totalization”, which is defined to be the homotopy limit of the cosimplicial diag... |

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Citation Context ...rational homotopy theory In this section we review some well-known concepts and results from rational homotopy theory. The definitive reference for this subject (at least for the unstable version) is =-=[9]-=-. We also prove some folklore results for which we could not find a reference. Fix a field k of characteristic zero. We will need the following categories and functors: • DGL (and DGL1), the category ... |

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Citation Context ... is formal when its rational homotopy type is determined by its rational cohomology algebra, and it is coformal when its rational homotopy type is determined by its rational homotopy Lie algebra (see =-=[16]-=-). The first technical difficulty we will encounter is in properly defining a coformal operad in the category of spaces (such as the little d-disks operad). The reason this is not straightforward is t... |

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Citation Context .... This spectral sequence converges to πq+p(Emb(R, R d )) ⊗ Q, again when d ≥ 4 [20, Theorem 7.1]. Since rational homotopy groups of configuration spaces are well understood thanks to Cohen and Gitler =-=[6]-=-, the E 1 page of QπBKSS can be easily computed. The coface maps in K • are also well understood [18, 20] so it is not hard to write explicit formulas for the differential d 1 . Some low degree calcul... |

14 |
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Citation Context ...s are projection maps which forget certain points. The existence of this cosimplicial space was originally suggested by Goodwillie ([11, Example 5.1.4]) and was subsequently proved by Sinha ([21] and =-=[20]-=-; see Section 4 in this paper for more details). It follows that for any abelian group k, there is an associated homology Bousfield-Kan spectral sequence with coefficients in k (kHBKSS for short) with... |

14 |
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Citation Context ...rational homotopy theory In this section we review some well-known concepts and results from rational homotopy theory. The definitive reference for this subject (at least for the unstable version) is =-=[7]-=-. We also prove some folklore results for which we could not find a reference. Fix a field k of characteristic zero. We will need the following categories and functors: • DGL (and DGL1), the category ... |

13 | A one-dimensional embedding complex
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Citation Context ...nce rational homotopy groups of configuration spaces are well understood thanks to Cohen and Gitler [6], the E 1 page of QπBKSS can be easily computed. The coface maps in K • are also well understood =-=[18, 20]-=- so it is not hard to write explicit formulas for the differential d 1 . Some low degree calculations in the E 2 page were carried out in [18]. The main result of this paper is the following Theorem 1... |

6 |
On the homology of the spaces of long knots
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Citation Context ...VICTOR TURCHIN, AND ISMAR VOLI Ć and a differential d 1 that can easily be made explicit [20]. This spectral sequence converges to Hq+p(Emb(R, R d );k) for d ≥ 4 [21, Theorem 7.2]. The main result of =-=[13]-=- says that this spectral sequence collapses at the E 2 page when k = Q. The goal of the present paper is to prove an analogous result for homotopy groups. Thus we consider the rational homotopy Bousfi... |

4 | Calculus of functors, operad formality, and rational homology of embedding spaces
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(Show Context)
Citation Context ... This line of argument is summarized in Theorem 4.1 which gives a somewhat specialized criterion for the collapse of a QπBKSS . A possible extension of our present results is suggested by the work in =-=[1]-=-. In that paper, it was proved that relative formality of the little disks operad implies the collapse of a certain spectral sequence computing the rational homology of the embedding space, Emb(M, R d... |

4 |
I Volić, Calculus of functors, operad formality, and rational homology of embedding spaces
- Arone, Lambrechts
(Show Context)
Citation Context ...cause the QπBKSS of a coformal cosimplicial space collapses at the E 2 page (Corollary 3.8).COFORMALITY AND π∗ OF LONG KNOTS 3 A possible extension of our present results is suggested by the work in =-=[1]-=-. In that paper, it was proved that relative formality of the little disks operad implies the collapse of a certain spectral sequence computing the rational homology of the embedding space, Emb(M, R d... |

4 |
Ismar Volić, The rational homology of spaces of long knots in codimension
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(Show Context)
Citation Context ...K • , with E 1 p,q = Hq(K −p ;k) and a differential d 1 that can easily be made explicit [19]. This spectral sequence converges to Hq+p(Emb(R, R d );k) for d ≥ 4 [20, Theorem 7.2]. The main result of =-=[11]-=- says that this spectral sequence collapses at the E 2 page when k = Q. The goal of the present paper is to prove an analogous result for homotopy groups. Thus we consider the rational homotopy Bousfi... |

2 | Formality of the little d-discs operad. In preparation. Available at http://palmer.wellesley.edu/˜ivolic/pages/papers.html - Lambrechts, Volić |

1 |
Box 400137 Charlottesville, VA 22904E-mail address: zga2m@virginia.edu URL: http://www.math.virginia.edu/~zga2m/ Institut Mathématique, Université Catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvainla-Neuve, Belgium E-mail address: pascal.lambrec
- Soc
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Citation Context ... into R d , and we will denote this space by Imm(R, R d ). In this paper, we consider the homotopy fiber Emb(R, R d ) := hofiber(Emb(R, R d ) ֒→ Imm(R, R d )). This space was studied, for example, in =-=[21]-=- (where it was denoted by Ed). It is known (see [21, Proposition 5.17] for example) that Emb(R, R d ) ≃ Emb(R, R d ) × Ω 2 S d−1 , so that any information about the homotopy type of Emb(R, R d ) trans... |

1 |
and Ismar Volić. Formality of the little d-discs operad. Draft available at http://milnor.math.ucl.ac.be/plwiki/PascalLambrechtsProfessional/ListOfPublications
- Lambrechts
(Show Context)
Citation Context ...t strictly true. Morally speaking, this is a consequence of the fact that K • is homotopy equivalent to the little d-disks operad, which is formal by a theorem of Kontsevich [10, Theorem 2] (see also =-=[12]-=-). It is easy to show that the QHBKSS of a formal cosimplicial space collapses at the E 2 page (see Proposition 3.6). However, the authors of [11] were unable to prove formality of K • . Instead, they... |

1 |
The topology of spaces of knots. Submitted. arXiv: math.AT/0202287, version 4
- Sinha
- 2004
(Show Context)
Citation Context ...maps are projection maps forgetting certain points. The existence of this cosimplicial space was originally suggested by Goodwillie ([9, Example 5.1.4]) and was subsequently proved by Sinha ([20] and =-=[19]-=-; see Section 4 for more details). 1991 Mathematics Subject Classification. Primary: 57Q45; Secondary: 55P62, 55P48. Key words and phrases. knot spaces, embedding calculus, formality, operads, Bousfie... |

1 |
E-mail address: zga2m@virginia.edu Institut Mathématique, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium E-mail address: lambrechts@math.ucl.ac.be URL: http://milnor.math.ucl.ac.be/plwiki
- Operads, Soc
- 2006
(Show Context)
Citation Context ... of R into R d , and we will denote this space by Imm(R, R d ). In this paper, we study the homotopy fiber Emb(R, R d ) := hofiber(Emb(R, R d ) ֒→ Imm(R, R d )). This space was studied for example in =-=[20]-=- (where it was denoted by Ed). It is known (for example, see [20, Proposition 5.17]) that Emb(R, R d ) ≃ Emb(R, R d ) × Ω 2 S d−1 , thus any information about the homotopy type of Emb(R, R d ) transla... |