Rational homology and homotopy of high-dimensional string links
-
Paul Arnaud Songhafouo Tsopméné
and
Victor Turchin
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
Funding statement: 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).
Acknowledgements
The second author thanks S. Melikhov, A. Skopenkov, and M. Skopenkov for communication.
References
[1] V. I. Arnol’d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231. Search in 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. 10.2140/gt.2014.18.1261Search in Google Scholar
[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. 10.5802/aif.2924Search in Google Scholar
[4] D. Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995), no. 1, 13–32. 10.1142/S021821659500003XSearch in Google Scholar
[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. 10.1112/topo.12048Search in Google Scholar
[6] A. J. Casson, Link cobordism and Milnor’s invariant, Bull. Lond. Math. Soc. 7 (1975), 39–40. 10.1112/blms/7.1.39Search in Google Scholar
[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. 10.1142/S0218216505004019Search in Google Scholar
[8] S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University Press, Cambridge, 2012. 10.1017/CBO9781139107846Search in 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. 10.1090/S0002-9939-2014-12331-8Search in 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. 10.1007/BFb0080464Search in Google Scholar
[11] J. Conant, M. Kassabov and K. Vogtmann, Higher hairy graph homology, Geom. Dedicata 176 (2015), 345–374. 10.1007/s10711-014-9972-4Search in Google Scholar
[12] J. Conant, R. Schneiderman and P. Teichner, Tree homology and a conjecture of Levine, Geom. Topol. 16 (2012), no. 1, 555–600. 10.2140/gt.2012.16.555Search in Google Scholar
[13]
D. Crowley, S. C. Ferry and M. Skopenkov,
The rational classification of links of codimension
[14] J. Ducoulombier and V. Turchin, Delooping the manifold calculus tower for closed discs, preprint (2017), https://arxiv.org/abs/1708.02203. Search in Google Scholar
[15] W. Dwyer and K. Hess, Long knots and maps between operads, Geom. Topol. 16 (2012), no. 2, 919–955. 10.2140/gt.2012.16.919Search in Google Scholar
[16] W. Dwyer and K. Hess, A delooping of the space of string links, preprint (2015), https://arxiv.org/abs/1501.00575. Search in Google Scholar
[17] W. Dwyer and K. Hess, Delooping the space of long embeddings, to appear. Search in Google Scholar
[18] B. Fresse, Modules Over Operads and Functors, Lecture Notes in Math. 1967, Springer, Berlin, 2009. 10.1007/978-3-540-89056-0Search in Google Scholar
[19]
B. Fresse, V. Turchin and T. Willwacher,
The rational homotopy of mapping spaces of
[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. Search in Google Scholar
[21] E. Getzler and M. M. Kapranov, Modular operads, Compos. Math. 110 (1998), no. 1, 65–126. 10.1023/A:1000245600345Search in Google Scholar
[22] T. G. Goodwillie and J. R. Klein, Multiple disjunction for spaces of smooth embeddings, J. Topol. 8 (2015), no. 3, 651–674. 10.1112/jtopol/jtv008Search in Google Scholar
[23] T. G. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory. II, Geom. Topol. 3 (1999), 103–118. 10.2140/gt.1999.3.103Search in Google Scholar
[24] N. Habegger and G. Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), no. 6, 1253–1289. 10.1016/S0040-9383(99)00041-5Search in Google Scholar
[25] N. Habegger and W. Pitsch, Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications 12 (2003), no. 3, 333–345. 10.1142/S0218216503002494Search in Google Scholar
[26]
A. Haefliger,
Knotted
[27] A. Haefliger, Enlacements de sphères en codimension supérieure à 2, Comment. Math. Helv. 41 (1966/1967), 51–72. 10.1007/BF02566868Search in Google Scholar
[28] V. Hinich and A. Vaintrob, Cyclic operads and algebra of chord diagrams, Selecta Math. (N.S.) 8 (2002), no. 2, 237–282. 10.1007/s00029-002-8106-2Search in Google Scholar
[29] M. Hovey, Model Categories, Math. Surveys Monogr. 63, American Mathematical Society, Providence, 1999. Search in Google Scholar
[30] U. Koschorke, A generalization of Milnor’s μ-invariants to higher-dimensional link maps, Topology 36 (1997), no. 2, 301–324. 10.1016/0040-9383(96)00018-3Search in Google Scholar
[31] P. Lambrechts and I. Volić, Formality of the little N-disks operad, Mem. Amer. Math. Soc. 1079 (2014), 1–116. Search in 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. 10.2140/agt.2002.2.1197Search in Google Scholar
[33] J.-L. Loday and B. Vallette, Algebraic Operads, Grundlehren Math. Wiss. 346, Springer, Heidelberg, 2012. 10.1007/978-3-642-30362-3Search in Google Scholar
[34]
R. Longoni,
Nontrivial classes in
[35] M. Markl, Loop homotopy algebras in closed string field theory, Comm. Math. Phys. 221 (2001), no. 2, 367–384. 10.1007/PL00005575Search in Google Scholar
[36] B. A. Munson, Derivatives of the identity and generalizations of Milnor’s invariants, J. Topol. 4 (2011), no. 2, 383–405. 10.1112/jtopol/jtr005Search in Google Scholar
[37] B. A. Munson and I. Volić, Multivariable manifold calculus of functors, Forum Math. 24 (2012), no. 5, 1023–1066. 10.1515/form.2011.095Search in 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. 10.1007/s40062-013-0033-8Search in Google Scholar
[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. 10.1090/conm/682/13809Search in Google Scholar
[40] T. Pirashvili, Dold-Kan type theorem for Γ-groups, Math. Ann. 318 (2000), no. 2, 277–298. 10.1007/s002080000120Search in Google Scholar
[41] T. Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 151–179. 10.1016/S0012-9593(00)00107-5Search in Google Scholar
[42] K. Sakai, Configuration space integrals for embedding spaces and the Haefliger invariant, J. Knot Theory Ramifications 19 (2010), no. 12, 1597–1644. 10.1142/S0218216510008583Search in Google Scholar
[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. 10.1017/S0305004111000429Search in Google Scholar
[44] D. Sinha, A pairing between graphs and trees, preprint (2005), https://arxiv.org/abs/math/0502547. Search in Google Scholar
[45] P. A. Songhafouo Tsopméné, The rational homology of spaces of long links, Algebr. Geom. Topol. 16 (2016), no. 2, 757–782. 10.2140/agt.2016.16.757Search in Google Scholar
[46]
P. A. Songhafouo Tsopméné and V. Turchin,
Euler characteristics for spaces of string links and the modular envelope of
[47] V. Turchin, Hodge-type decomposition in the homology of long knots, J. Topol. 3 (2010), no. 3, 487–534. 10.1112/jtopol/jtq015Search in Google Scholar
[48] V. Turchin, Delooping totalization of a multiplicative operad, J. Homotopy Relat. Struct. 9 (2014), no. 2, 349–418. 10.1007/s40062-013-0032-9Search in Google Scholar
[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. 10.1353/ajm.2018.0006Search in Google Scholar
[50]
V. Turchin and T. Willwacher,
Hochschild–Pirashvili homology on suspensions and representations of
[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. Search in Google Scholar
[52] M. Weiss, Embeddings from the point of view of immersion theory. I, Geom. Topol. 3 (1999), 67–101. 10.2140/gt.1999.3.67Search in Google Scholar
[53] M. S. Weiss, Homology of spaces of smooth embeddings, Q. J. Math. 55 (2004), no. 4, 499–504. 10.1093/qmath/hah009Search in Google Scholar
[54] S. Whitehouse, Gamma homology of commutative algebras and some related representations of the symmetric group, PhD Thesis, Warwick University, 1994. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Upper bounds for geodesic periods over rank one locally symmetric spaces
- Additive bases with coefficients of newforms
- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
- On the non-existence of the Mackey topology for locally quasi-convex groups
- The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
- The cup product of Brooks quasimorphisms
- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
- Rational homology and homotopy of high-dimensional string links
- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
- Contractibility of the stability manifold for silting-discrete algebras
- Order of the canonical vector bundle over configuration spaces of spheres
- An endpoint version of uniform Sobolev inequalities
- Space-time L2 estimates, regularity and almost global existence for elastic waves
- k-spaces and duals of non-archimedean metrizable locally convex spaces
- Path homology theory of multigraphs and quivers
Articles in the same Issue
- Frontmatter
- Upper bounds for geodesic periods over rank one locally symmetric spaces
- Additive bases with coefficients of newforms
- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
- On the non-existence of the Mackey topology for locally quasi-convex groups
- The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
- The cup product of Brooks quasimorphisms
- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
- Rational homology and homotopy of high-dimensional string links
- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
- Contractibility of the stability manifold for silting-discrete algebras
- Order of the canonical vector bundle over configuration spaces of spheres
- An endpoint version of uniform Sobolev inequalities
- Space-time L2 estimates, regularity and almost global existence for elastic waves
- k-spaces and duals of non-archimedean metrizable locally convex spaces
- Path homology theory of multigraphs and quivers
