Show Summary Details
More options …

# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 30, Issue 5

# Rational homology and homotopy of high-dimensional string links

Paul Arnaud Songhafouo Tsopméné
• Corresponding author
• Department of Mathematics and Statistics, University of Regina, 3737 Wascana Pkwy, Regina, SK, S4S 0A2, Canada
• Email
• Other articles by this author:
/ Victor Turchin
Published Online: 2018-03-08 | DOI: https://doi.org/10.1515/forum-2016-0192

## Abstract

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of ${\pi }_{0}$.

Keywords: String links; Goodwillie calculus; formality

MSC 2010: 57Q45

## References

• [1]

V. I. Arnol’d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231. Google Scholar

• [2]

G. Arone and V. Turchin, On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol. 18 (2014), no. 3, 1261–1322.

• [3]

G. Arone and V. Turchin, Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 1, 1–62.

• [4]

D. Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995), no. 1, 13–32.

• [5]

P. Boavida de Brito and M. Weiss, Spaces of smooth embeddings and configuration categories, preprint (2015), https://arxiv.org/abs/1502.01640; to appear in J. Topol.

• [6]

A. J. Casson, Link cobordism and Milnor’s invariant, Bull. Lond. Math. Soc. 7 (1975), 39–40.

• [7]

A. S. Cattaneo, P. Cotta-Ramusino and R. Longoni, Algebraic structures on graph cohomology, J. Knot Theory Ramifications 14 (2005), no. 5, 627–640.

• [8]

S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University Press, Cambridge, 2012. Google Scholar

• [9]

F. R. Cohen, R. Komendarczyk and C. Shonkwiler, Homotopy Brunnian links and the κ-invariant, Proc. Amer. Math. Soc. 143 (2015), no. 3, 1347–1362. Google Scholar

• [10]

F. R. Cohen, T. J. Lada and J. P. May, The Homology of Iterated Loop Spaces, Lecture Notes in Math. 533, Springer, Berlin, 1976. Google Scholar

• [11]

J. Conant, M. Kassabov and K. Vogtmann, Higher hairy graph homology, Geom. Dedicata 176 (2015), 345–374.

• [12]

J. Conant, R. Schneiderman and P. Teichner, Tree homology and a conjecture of Levine, Geom. Topol. 16 (2012), no. 1, 555–600.

• [13]

D. Crowley, S. C. Ferry and M. Skopenkov, The rational classification of links of codimension , Forum Math. 26 (2014), no. 1, 239–269. Google Scholar

• [14]

J. Ducoulombier and V. Turchin, Delooping the manifold calculus tower for closed discs, preprint (2017), https://arxiv.org/abs/1708.02203.

• [15]

W. Dwyer and K. Hess, Long knots and maps between operads, Geom. Topol. 16 (2012), no. 2, 919–955.

• [16]

W. Dwyer and K. Hess, A delooping of the space of string links, preprint (2015), https://arxiv.org/abs/1501.00575.

• [17]

W. Dwyer and K. Hess, Delooping the space of long embeddings, to appear.

• [18]

B. Fresse, Modules Over Operads and Functors, Lecture Notes in Math. 1967, Springer, Berlin, 2009. Google Scholar

• [19]

B. Fresse, V. Turchin and T. Willwacher, The rational homotopy of mapping spaces of ${E}_{n}$ operads, preprint (2017), https://arxiv.org/abs/1703.06123.

• [20]

E. Getzler and M. Kapranov, Cyclic operads and cyclic homology, Geometry, Topology and Physics for Raoul Bott, Conf. Proc. Lecture Notes Geom. Topol. 4, International Press, Cambridge (1995), 167–201. Google Scholar

• [21]

E. Getzler and M. M. Kapranov, Modular operads, Compos. Math. 110 (1998), no. 1, 65–126.

• [22]

T. G. Goodwillie and J. R. Klein, Multiple disjunction for spaces of smooth embeddings, J. Topol. 8 (2015), no. 3, 651–674.

• [23]

T. G. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory. II, Geom. Topol. 3 (1999), 103–118.

• [24]

N. Habegger and G. Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), no. 6, 1253–1289.

• [25]

N. Habegger and W. Pitsch, Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications 12 (2003), no. 3, 333–345.

• [26]

A. Haefliger, Knotted $\left(4k-1\right)$-spheres in $6k$-space, Ann. of Math. (2) 75 (1962), 452–466. Google Scholar

• [27]

A. Haefliger, Enlacements de sphères en codimension supérieure à 2, Comment. Math. Helv. 41 (1966/1967), 51–72.

• [28]

V. Hinich and A. Vaintrob, Cyclic operads and algebra of chord diagrams, Selecta Math. (N.S.) 8 (2002), no. 2, 237–282.

• [29]

M. Hovey, Model Categories, Math. Surveys Monogr. 63, American Mathematical Society, Providence, 1999. Google Scholar

• [30]

U. Koschorke, A generalization of Milnor’s μ-invariants to higher-dimensional link maps, Topology 36 (1997), no. 2, 301–324.

• [31]

P. Lambrechts and I. Volić, Formality of the little N-disks operad, Mem. Amer. Math. Soc. 1079 (2014), 1–116. Google Scholar

• [32]

J. Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group” [Algebr. Geom. Topol. 1 (2001), 243–270], Algebr. Geom. Topol. 2 (2002), 1197–1204.

• [33]

J.-L. Loday and B. Vallette, Algebraic Operads, Grundlehren Math. Wiss. 346, Springer, Heidelberg, 2012. Google Scholar

• [34]

R. Longoni, Nontrivial classes in ${H}^{*}\left(\mathrm{Imb}\left({S}^{1},{ℝ}^{n}\right)\right)$ from nontrivalent graph cocycles, Int. J. Geom. Methods Mod. Phys. 1 (2004), no. 5, 639–650. Google Scholar

• [35]

M. Markl, Loop homotopy algebras in closed string field theory, Comm. Math. Phys. 221 (2001), no. 2, 367–384.

• [36]

B. A. Munson, Derivatives of the identity and generalizations of Milnor’s invariants, J. Topol. 4 (2011), no. 2, 383–405.

• [37]

B. A. Munson and I. Volić, Multivariable manifold calculus of functors, Forum Math. 24 (2012), no. 5, 1023–1066. Google Scholar

• [38]

B. A. Munson and I. Volić, Cosimplicial models for spaces of links, J. Homotopy Relat. Struct. 9 (2014), no. 2, 419–454.

• [39]

K. E. Pelatt and D. P. Sinha, A geometric homology representative in the space of knots, Manifolds and K-Theory, Contemp. Math. 682, American Mathematical Society, Providence (2017), 167–188. Google Scholar

• [40]

T. Pirashvili, Dold-Kan type theorem for Γ-groups, Math. Ann. 318 (2000), no. 2, 277–298.

• [41]

T. Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 151–179.

• [42]

K. Sakai, Configuration space integrals for embedding spaces and the Haefliger invariant, J. Knot Theory Ramifications 19 (2010), no. 12, 1597–1644.

• [43]

K. Sakai and T. Watanabe, 1-loop graphs and configuration space integral for embedding spaces, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 3, 497–533.

• [44]

D. Sinha, A pairing between graphs and trees, preprint (2005), https://arxiv.org/abs/math/0502547.

• [45]

P. A. Songhafouo Tsopméné, The rational homology of spaces of long links, Algebr. Geom. Topol. 16 (2016), no. 2, 757–782.

• [46]

P. A. Songhafouo Tsopméné and V. Turchin, Euler characteristics for spaces of string links and the modular envelope of ${\mathcal{ℒ}}_{\mathrm{\infty }}$, preprint (2016), https://arxiv.org/abs/1609.00778; to appear in Homology Homotopy Appl.

• [47]

V. Turchin, Hodge-type decomposition in the homology of long knots, J. Topol. 3 (2010), no. 3, 487–534.

• [48]

V. Turchin, Delooping totalization of a multiplicative operad, J. Homotopy Relat. Struct. 9 (2014), no. 2, 349–418.

• [49]

V. Turchin and T. Willwacher, Relative (non-)formality of the little cubes operads and the algebraic Cerf lemma, preprint (2014), https://arxiv.org/abs/1409.0163; to appear in Amer. J. Math.

• [50]

V. Turchin and T. Willwacher, Hochschild–Pirashvili homology on suspensions and representations of $\mathrm{Out}\left({F}_{n}\right)$, preprint (2015), https://arxiv.org/abs/1507.08483; to appear in Ann. Sci. Éc. Norm. Supér.

• [51]

I. Volić, On the cohomology of spaces of links and braids via configuration space integrals, Sarajevo J. Math. 6(19) (2010), no. 2, 241–263. Google Scholar

• [52]

M. Weiss, Embeddings from the point of view of immersion theory. I, Geom. Topol. 3 (1999), 67–101.

• [53]

M. S. Weiss, Homology of spaces of smooth embeddings, Q. J. Math. 55 (2004), no. 4, 499–504.

• [54]

S. Whitehouse, Gamma homology of commutative algebras and some related representations of the symmetric group, PhD Thesis, Warwick University, 1994. Google Scholar

Revised: 2017-09-27

Published Online: 2018-03-08

Published in Print: 2018-09-01

This work has been supported by Fonds de la Recherche Scientifique-FNRS (F.R.S.-FNRS). It has also been supported by the Kansas State University (KSU), where this paper was partially written during the stay of the first author, and which he thanks for hospitality. The second author is partially supported by the Simons Foundation “Collaboration grant for mathematicians” (award ID: 519474).

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1209–1235, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.