Quandle cohomology is a Quillen cohomology
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- by Markus Szymik PDF
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Abstract:
Racks and quandles are fundamental algebraic structures related to the topology of knots, braids, and the Yang–Baxter equation. We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. This makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change).References
- Nicolás Andruskiewitsch and Matías Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–243. MR 1994219, DOI 10.1016/S0001-8708(02)00071-3
- R. Bott and H. Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1953), 320–337 (1954). MR 60233, DOI 10.1007/BF02564566
- J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989. MR 1990571, DOI 10.1090/S0002-9947-03-03046-0
- Frans Clauwens, The algebra of rack and quandle cohomology, J. Knot Theory Ramifications 20 (2011), no. 11, 1487–1535. MR 2854230, DOI 10.1142/S0218216511010073
- Albrecht Dold and Dieter Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201–312 (German, with French summary). MR 150183
- Michael Eisermann, Homological characterization of the unknot, J. Pure Appl. Algebra 177 (2003), no. 2, 131–157. MR 1954330, DOI 10.1016/S0022-4049(02)00068-3
- P. Etingof and M. Graña, On rack cohomology, J. Pure Appl. Algebra 177 (2003), no. 1, 49–59. MR 1948837, DOI 10.1016/S0022-4049(02)00159-7
- M. Farinati, J. A. Guccione, and J. J. Guccione, The homology of free racks and quandles, Comm. Algebra 42 (2014), no. 8, 3593–3606. MR 3196064, DOI 10.1080/00927872.2013.790392
- Roger Fenn, Colin Rourke, and Brian Sanderson, An introduction to species and the rack space, Topics in knot theory (Erzurum, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 399, Kluwer Acad. Publ., Dordrecht, 1993, pp. 33–55. MR 1257904
- Roger Fenn, Colin Rourke, and Brian Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356. MR 1364012, DOI 10.1007/BF00872903
- Roger Fenn, Colin Rourke, and Brian Sanderson, James bundles, Proc. London Math. Soc. (3) 89 (2004), no. 1, 217–240. MR 2063665, DOI 10.1112/S0024611504014674
- Roger Fenn, Colin Rourke, and Brian Sanderson, The rack space, Trans. Amer. Math. Soc. 359 (2007), no. 2, 701–740. MR 2255194, DOI 10.1090/S0002-9947-06-03912-2
- Martin Frankland, Behavior of Quillen (co)homology with respect to adjunctions, Homology Homotopy Appl. 17 (2015), no. 1, 67–109. MR 3338541, DOI 10.4310/HHA.2015.v17.n1.a3
- M. A. Hill, M. J. Hopkins, and D. C. Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016), no. 1, 1–262. MR 3505179, DOI 10.4007/annals.2016.184.1.1
- Nicholas Jackson, Extensions of racks and quandles, Homology Homotopy Appl. 7 (2005), no. 1, 151–167. MR 2155522
- David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65. MR 638121, DOI 10.1016/0022-4049(82)90077-9
- Daniel M. Kan, On c.s.s. categories, Bol. Soc. Mat. Mexicana 2 (1957), 82–94. MR 0096211
- F. William Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869–872. MR 158921, DOI 10.1073/pnas.50.5.869
- R. A. Litherland and Sam Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178 (2003), no. 2, 187–202. MR 1952425, DOI 10.1016/S0022-4049(02)00211-6
- S. V. Matveev, Distributive groupoids in knot theory, Math. USSR 47 (1984) 73–83.
- J. C. Moore, Homotopie des complexes monoïdaux, I, Séminaire Henri Cartan, 1955, Exp. 18, 1–8.
- M. Niebrzydowski and J. H. Przytycki, Homology operations on homology of quandles, J. Algebra 324 (2010), no. 7, 1529–1548. MR 2673749, DOI 10.1016/j.jalgebra.2010.04.033
- Takefumi Nosaka, On homotopy groups of quandle spaces and the quandle homotopy invariant of links, Topology Appl. 158 (2011), no. 8, 996–1011. MR 2786669, DOI 10.1016/j.topol.2011.02.006
- Takefumi Nosaka, On quandle homology groups of Alexander quandles of prime order, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3413–3436. MR 3042590, DOI 10.1090/S0002-9947-2013-05754-6
- Takefumi Nosaka, Quandles and topological pairs, SpringerBriefs in Mathematics, Springer, Singapore, 2017. Symmetry, knots, and cohomology. MR 3729413, DOI 10.1007/978-981-10-6793-8
- Józef H. Przytycki and Krzysztof K. Putyra, The degenerate distributive complex is degenerate, Eur. J. Math. 2 (2016), no. 4, 993–1012. MR 3572555, DOI 10.1007/s40879-016-0116-2
- Józef H. Przytycki and Seung Yeop Yang, The torsion of a finite quasigroup quandle is annihilated by its order, J. Pure Appl. Algebra 219 (2015), no. 10, 4782–4791. MR 3346517, DOI 10.1016/j.jpaa.2015.03.013
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
- Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068
- D. G. Quillen, Homology of commutative rings, mimeographed notes, MIT, 1968.
- Charles Rezk, Every homotopy theory of simplicial algebras admits a proper model, Topology Appl. 119 (2002), no. 1, 65–94. MR 1881711, DOI 10.1016/S0166-8641(01)00057-8
- Markus Szymik, Permutations, power operations, and the center of the category of racks, Comm. Algebra 46 (2018), no. 1, 230–240. MR 3764859, DOI 10.1080/00927872.2017.1316857
- M. Szymik, Alexander–Beck modules detect the unknot, http://arxiv.org/abs/1610.08306 (2016); Fund. Math. (to appear).
Additional Information
- Markus Szymik
- Affiliation: Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, 7491 Trondheim, Norway
- MR Author ID: 816144
- Email: markus.szymik@ntnu.no
- Received by editor(s): January 17, 2017
- Received by editor(s) in revised form: January 25, 2018
- Published electronically: October 24, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5823-5839
- MSC (2010): Primary 18G50, 57M27; Secondary 18C10, 20N02, 55U35
- DOI: https://doi.org/10.1090/tran/7616
- MathSciNet review: 3937311