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Various Topics in Rack and Quandle Homology Various topics in rack and quandle homology April 14, 2010 Master's thesis in Mathematics Jorik Mandemaker supervisor: Dr. F.J.-B.J. Clauwens second reader: Prof. Dr. F. Keune student number: 0314145 Faculty of Science (FNWI) Radboud University Nijmegen Contents 1 Introduction 5 1.1 Racks and quandles . 6 1.2 Quandles and knots . 7 1.3 Augmented quandles and the adjoint group . 8 1.3.1 The adjoint group . 9 1.4 Connected components . 9 1.5 Rack and quandle homology . 10 1.6 Quandle coverings . 10 1.6.1 Extensions . 11 1.6.2 The fundamental group of a quandle . 12 1.7 Historical notes . 12 2 The second cohomology group of alexander quandles 14 2.1 The adjoint group of Alexander quandles . 15 2.1.1 Connected Alexander quandles . 15 2.1.2 Alexander quandles with connected components . 18 2.2 The second cohomology group of connected Alexander quandles . 19 3 Polynomial cocycles 22 3.1 Introduction . 23 3.2 Polynomial cochains . 23 3.3 A decomposition and filtration of the complex . 24 3.4 The 3-cocycles . 25 3.5 Variable Reduction . 27 3.6 Proof of surjectivity . 28 3.6.1 t = s .............................. 29 3.6.2 t < s .............................. 30 3.7 Final steps . 33 4 Rack and quandle modules 35 4.1 Rack and Quandle Modules . 36 4.1.1 Definitions . 36 4.1.2 Some examples . 38 4.2 Reduced rack modules . 39 1 4.3 Right modules and tensor products . 40 4.4 The rack algebra . 42 4.5 Examples . 44 4.5.1 A free resolution for trivial quandles . 44 4.5.2 R3 ............................... 44 5 Trunks 46 5.1 Introduction . 47 5.2 Trunks . 47 5.3 -sets and their realizations . 48 5.4 The nerve of a trunk . 49 6 Homology operators 51 6.1 The augmented rack space . 52 6.1.1 H1(BInnX) .......................... 54 6.2 Homology operators of negative degree . 55 0 6.2.1 H (BGX X).......................... 58 6.3 Variations . 59 6.3.1 A different G ......................... 59 6.3.2 Quandle homology . 59 6.4 Relation to the literature . 60 6.4.1 Some computer calculations . 60 6.5 Some other operators . 61 7 The cup product of the rack space 63 7.1 The cup product of the realization of a -set . 64 7.2 A triangulation of the realization of a -set . 65 7.3 The cup product . 67 2 Summary The goal of this thesis project was originally to survey the current literature on racks and quandles in order to comment on it or extend it. Eventually this resulted in this thesis being a mix of various loosely related topics. Since rack and quandle theory is a rather obscure part of mathematics this thesis contains a general introduction to racks and quandles covering all the ba- sics. The first chapter contains this introduction and can generally be regarded as "common" knowledge about quandles. The second and third chapter are both concerned with computing the co- homology of certain Alexander quandles. The former being an application of the work by Eisermann [7]. The calculation in 2.1.1 I owe to my supervisor F. Clauwens. But the rest of the chapter is my own work. The third chapter is a correction of the work by Mochizuki [17, 18]. In these articles Mochizuki claims to have found a basis for the third cohomology group of some Alexander quandles. However this list contains a coboundary, so it is clearly incorrect. The next chapter focuses on the rack and quandle modules from the work by Jackson [12, 13]. Jackson studied these modules as an alternate way of defining rack and quandle homology. For connected quandles we'll show that those modules coincide with modules over a subring of the group ring of the adjoint group of the quandle. Chapter 5 is another introductionary chapter which serves mostly to set the terminology used in the final two chapters. It covers the basics of trunks and their classifying spaces as found in [9]. Chapter 6 explores the homology of the augmented rack space which proves a useful tool to study the homology of racks. It also shines new light on some of the work by Niebrzydowski and Przytycki [19]. The orginal idea that the augmented rack spaces is a topological monoid that produces homology operators is once again due to F. Clauwens. Section 6.1.1 and further is my own work with the exception of formula 6.6. The final chapter studies the yet unexplored the cup product on the coho- mology of -sets which in particular covers that of racks and quandles. 3 Acknowledgements I would like to thank my thesis supervisor Frans Clauwens who was very patient with me over the course of this thesis project. He worked tirelessly on racks and quandles himself. He often pointed me in the right direction and explained many interesting things to me. In particular I have to thank him for the computation of chapter 2.1.1 and the insight that the cycles and cocycles of BGX produce homology operators (chapter 6). I would also like to thank Frans Keune for being the second reader of my thesis. 4 Chapter 1 Introduction 5 1.1 Racks and quandles Definition 1.1.1. A quandle (Q; B; C) is a set Q with two binary operators B and C such that the following axioms hold. (Q1) a B a = a for all a in Q (Q2) (a C b) B b = a = (a B b) C b for all a; b in Q (Q3) (a B b) B c = (a B c) B (b B c) for all a; b; c in Q A set R with two binary operators satisfying axioms (Q2) and (Q3) but not necessarily (Q1) is called a rack. If (Q; B; C) is a quandle or rack than the operation C is uniquely determined by B. Because of this we will usually omit it from the notation and simply write (Q; B) or even just Q. Definition 1.1.2. A quandle homomorphism between two quandles X and Y is a function φ : X ! Y such that φ(aBb) = φ(a)Bφ(b) and φ(aCb) = φ(a)Cφ(b) for all a; b 2 X. The composition of two quandle homomorphisms is again a quandle homomorphism so we can speak of the category Qnd with objects quandles and morphisms the quandle homomorphism. In the same way we obtain the category of racks denoted by Rack. Convention 1.1.3. We let both B and C associate to the left. So when we write a B b C c B d we mean ((a B b) C c) B d Theorem 1.1.1. If (Q; B; C) is a quandle then (Q; C; B) is also a quandle called the dual quandle. Definition 1.1.4. A quandle or rack Q is said to be involutive if it satisfies (QInv) a B b B b = a for all a; b in Q A quandle or rack Q is said to be abelian if it satisfies (Qab) (a B x) B (y B b) = (a B y) B (x B b) for all a; b; x; y in Q −1 Example 1.1.1. Every group G becomes a quandle by setting a B b = b ab −1 and a C b = bab . This quandle is called the conjugation quandle of G denoted by Conj(G). Every group homomorphism between two groups G and H is also a quandle homomorphism between Conj(G) and Conj(H). Thus we obtain a functor Conj : Grp ! Qnd. Example 1.1.2. If X is any set then X becomes a quandle by aBb = aCb = a called the trivial quandle on X. 6 Example 1.1.3. If M is an abelian group and T : M ! M is an automorphism of M. Then M becomes a quandle by setting x B y = T (x) + (1 − T )(y) −1 x C y = T (x − (1 − T )(y)) This quandle is called the Alexander quandle Alex(M; T ). Example 1.1.4. The dihedral quandle Rk is an Alexander quandle where the group is Zk and the automorphism is multiplication by −1. An alternate descrip- tion is that Rk is the subquandle of Conj(Dk) consisting of all the reflections. The automorphism group Aut(Q) of a quandle Q consists of all bijective quandle homomorphisms φ from Q to itself. We let this group act on Q from the right by evaluation notated by aφ = φ(a). This means that for φ, 2 Aut(Q) their composition φψ is given by a(φψ) = (aφ) . For every element a 2 Q we have an automorphism ρa defined by ρa(b) = b B a. Definition 1.1.5. The subgroup of Aut(Q) generated by the ρa is called the inner automorphism group of Q. This group is notated by Inn(Q). We also have the map inn : Q ! Inn(Q) defined by inn(a) = ρa. Convention 1.1.6. As a subgroup of Aut(Q) Inn(Q) acts on Q. We will often b inn(b) h −1 write a for a = a B b. We will also write g = h gh for conjugation in a group G in light of the quandle Conj(G). 1.2 Quandles and knots Quandles are closely related to knots. Every oriented knot has a quandle asso- ciated with it, called the fundamental quandle of the knot. It's easy to read of a knot diagram.
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