The homotopy types of free racks and quandles

We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor’s theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, more complicated than what any Wirtinger presentation coming from a diagram predicts.

[1]  The rack space , 2003, math/0304228.

[2]  Y. Berest.  Homotopical Algebra , 2019, Higher Categories and Homotopical Algebra.

[3]  C. Rourke,et al.  James bundles , 2003, math/0301354.

[4]  Carol Bult,et al.  PERMUTATIONS , 1994 .

[5]  F. W. Lawvere,et al.  FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Markus Szymik.  Quandle cohomology is a Quillen cohomology , 2016, Transactions of the American Mathematical Society.

[7]  Masahico Saito,et al.  Quandle cohomology and state-sum invariants of knotted curves and surfaces , 1999, math/9903135.

[8]  D. Anderson.  Convergent functors and spectra , 1974 .

[9]  David Joyce,et al.  A classifying invariant of knots, the knot quandle , 1982 .

[10]  O. Randal-Williams.  Homology of Hurwitz spaces and the Cohen--Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland) , 2019, 1906.07447.

[11]  J. Guccione,et al.  The Homology of Free Racks and Quandles , 2014 .

[12]  Travis Schedler,et al.  On set-theoretical solutions of the quantum Yang-Baxter equation , 1997 .

[13]  Bjørn Ian Dundas,et al.  The Local Structure of Algebraic K-Theory , 2012 .

[14]  Quandles and Monodromy , 2002, math/0205162.

[15]  V. Drinfeld.  On some unsolved problems in quantum group theory , 1992 .

[16]  M. Lydakis.  Smash products and Γ-spaces , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  D. Anderson.  Chain functors and homology theories , 1971 .

[18]  F. Clauwens.  The algebra of rack and quandle cohomology , 2010, 1004.4423.

[19]  Paul G. Goerss,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[20]  Nathan Geer,et al.  Holonomy braidings , biquandles and quantum invariants of links with SL 2 ( C ) flat connections , 2018 .

[21]  Markus Szymik.  Alexander–Beck modules detect the unknot , 2016, Fundamenta Mathematicae.

[22]  Roger Fenn,et al.  Trunks and classifying spaces , 1995, Appl. Categorical Struct..

[23]  George A. Elliott,et al.  K-theory , 1999 .

[24]  S. Matveev.  DISTRIBUTIVE GROUPOIDS IN KNOT THEORY , 1984 .

[25]  J. Adams,et al.  Algebraic Topology: A Student's Guide , 1972 .

[26]  Nicolas Andruskiewitsch,et al.  From racks to pointed Hopf algebras , 2002, math/0202084.

[27]  Symmetric spectra , 1998, math/9801077.

[28]  A. K. Bousfield,et al.  Homotopy theory of Γ-spaces, spectra, and bisimplicial sets , 1978 .

[29]  E. Brieskorn,et al.  Automorphic sets and braids and singularities , 1988 .

[30]  Takefumi Nosaka.  On the fundamental 3-classes of knot group representations , 2016, Geometriae Dedicata.

[31]  R. H. Crowell.  The group $G'/G''$ of a knot group $G$ , 1963 .

[32]  Patrick Dehornoy,et al.  Braids and self-distributivity , 2000 .

[33]  C. Rourke,et al.  HOMOLOGICAL CHARACTERIZATION OF THE UNKNOT , 2003 .

[34]  Roger Fenn,et al.  RACKS AND LINKS IN CODIMENSION TWO , 1992 .

[35]  C. Rourke,et al.  An Introduction to Species and the Rack Space , 1993 .

[36]  Akshay Venkatesh,et al.  Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II , 2009, 0912.0325.

[37]  V. Lebed,et al.  The homology of permutation racks , 2020, 2011.04524.

[38]  Markus Szymik.  Permutations, power operations, and the center of the category of racks , 2016, 1609.08687.

[39]  S. Schwede.  Stable homotopy of algebraic theories , 2001 .

[40]  G. Segal,et al.  Categories and cohomology theories , 1974 .

[41]  S. Schwede.  Stable homotopical algebra and Γ-spaces , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[42]  J. Przytycki,et al.  Homology operations on homology of quandles , 2009, 0907.4732.