## Homological Algebra of Racks and Quandles

Citations: | 1 - 1 self |

### BibTeX

@MISC{Jackson_homologicalalgebra,

author = {Nicholas James Jackson and Supervised Prof and Colin Rourke and Wilfred Leslie Jackson},

title = {Homological Algebra of Racks and Quandles},

year = {}

}

### OpenURL

### Abstract

Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expansions . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quandle extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Involutory extensions . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Modules 29 2.1 Rack modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Beck modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 A digression on G-modules . . . . . . . . . . . . . . . . . . . . . 41 2.4 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The rack

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Citation Context ...o axioms loosely analogous to the second and third Reidemeister moves. A quandle is a rack which satises a further axiom, itself loosely analogous to thesrst Reidemeister move. It is well-known [14] [=-=21-=-] [10] that such objects provide interesting collections of invariants of both classical links and higher-dimensional codimension-2 embedded manifolds. There is a well-dened notion of homology and coh... |

64 | Quandle cohomology and state-sum invariants of knotted curves and surfaces, toappear Trans AMS
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Citation Context ... hs : s 4 = 1i. The link 6 2 3 may be coloured with this quandle in sixteen ways, as depicted in a b f e c d Figure 7.4: The link 6 2 3 table 7.1. A tedious and routine calculation (similar to one in [10]) reveals that H 2 Q (D 4 ; A) = Z 4 Z 2 Z 4 . Let be the cocycle dened by (a; b) = 8 > > > > > > > > > > > > > > > > > > > > > > > > : s 2 A 1 = Z 4 if a = 1 and b = 2; 3 t 2 A 2 = Z 2 if a =... |

59 |
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Citation Context ...? f The Beck modules over G in the category Group are exactly the kernels of split extensions of G. This is also the case in the categories LieAlg, AssocAlg and CommRing; for details refer to [2] and =-=[25]-=-. It thus seems natural to suspect that the abelian group objects in the category Rack=X are exactly the split extensions examined in the previous chapter, and that the r^ole of coecient modules for r... |

43 | From racks to pointed Hopf algebras
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Citation Context ...eroth section of each chapter should be regarded as expository rather than original. Some aspects of the work in thesrst three chapters has been duplicated independently by Andruskiewitsch and Gra~na =-=[-=-1], although with many dierences in notation, motivation, and emphasis. I have not submitted any of this material in partial or complete fullment of the requirements for another degree at this or any ... |

37 |
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Citation Context ...re X is a group, @ is a group homomorphism, and the crossed module identity holds: a @b = b 1 ab for all a; b 2 X . Crossed modules weresrst studied, in the context of homotopy theory, by Whitehead [=-=29]-=-. More recently, their cohomology has been studied by Carrasco, Cegarra, and Grandjean [7], and by Paoli [24]. It would be interesting to see if a (co)homology theory for augmented racks agrees with c... |

34 | Triples, algebras and cohomology
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Citation Context ... the preceding discussion of extensions. It is shown that these objects (essentially the `extending' objects of the previous chapter) form an abelian category RModX and are exactly the `Beck modules' =-=[5]-=- [2] over X in the category Rack. Thus they are suitable coecient modules for (co)homology theories. The theory is further specialised to the subcategories Quandle, InvRack, and InvQuandle to obtain n... |

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Citation Context ...eories, while the formulations of the Kunneth formuland the universal coecient theorems are stated and proved in terms of derived functors, although analogous results hold for the original theories [1=-=1]-=-. The spectral sequence derives from a general theorem of Grothendieck pertaining to derived functors on module categories with enough injectives, and is hence not valid for the original topologically... |

20 |
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Citation Context ... is balanced; that is, every bimorphism (morphism which is both epic and monic) is an isomorphism. Curious readers are directed to any suitable book on category theory or homological algebra (such as =-=[2-=-3], [6], or [28]). 2.2 Beck modules The following result justies our use of the term `rack module' to denote the objects of RModX : Theorem 2.5 The category RModX of rack modules over a (xed) rack X i... |

19 | Some Results in Geometric Topology and Geometry
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Citation Context ...anifolds. There is a well-dened notion of homology and cohomology groups of a rack or quandle, dened in terms of an appropriate classifying space (the `rack space'); these have already been studied [1=-=8]-=- [17] [16], as has a related theory, that of `quandle (co)homology' [10]. My original objective was to generalise these existing theories and show how they could be regarded as derived functors constr... |

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Toposes, Triples, and Theories. Number 278 in Grundlehren der mathematischen Wissenschaften
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Citation Context ...struction known variously as a `cotriple', `standard construction', `comonad', or `cotriad'. The reader is directed to their work (especially Beck's doctoral thesis [5] and the book by Barr and Wells =-=[4]) for deta-=-iled proofs of the assertions and theory in this expository section. A cotriple (?; "; ) in a category C is an endofunctor ? : C ! C together with natural transformations " : ? ! Id C ands: ... |

15 |
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Citation Context ...l x; y 2 X . In the terminology of later chapters they are quandle extensions by a trivial homogeneous quandle module. Corollary 1.9 Carter, Saito and Elhamdadi's theory of twisted quandle extensions =-=[9]-=- is the special case of abelian quandle extensions where the coecient groups A = A x are all set to the same Alexander quotient Z[t; t 1 ]=h(t) where h(t) is a Laurent polynomial in one variable t, an... |

14 | Homology and standard constructions - Barr, Beck - 1969 |

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Citation Context ... b 1 ab for all a; b 2 X . Crossed modules weresrst studied, in the context of homotopy theory, by Whitehead [29]. More recently, their cohomology has been studied by Carrasco, Cegarra, and Grandjean =-=[7]-=-, and by Paoli [24]. It would be interesting to see if a (co)homology theory for augmented racks agrees with crossed-module (co)homology in the appropriate cases. Cotriple homology Another potentially... |

12 |
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Citation Context ...] by Fenn, Rourke, and Sanderson. 1.1 Extensions and expansions We begin by devising a suitable general theory of extensions for racks. Rack extensions have been studied before, in particular by Ryder=-=[27]-=- under the name `expansions'; the constructs which she dubs `extensions' are in some sense racks formed by disjoint unions, whereby the original rack becomes a subrack of the `extended' rack. Ryder's ... |

9 |
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Citation Context ...ng two axioms loosely analogous to the second and third Reidemeister moves. A quandle is a rack which satises a further axiom, itself loosely analogous to thesrst Reidemeister move. It is well-known [=-=14-=-] [21] [10] that such objects provide interesting collections of invariants of both classical links and higher-dimensional codimension-2 embedded manifolds. There is a well-dened notion of homology an... |

8 |
Trunks and classifying spaces. Applied categorical structures
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Citation Context ...re there is no ambiguity, we may refer to Trunk(C) as C; in particular we will adopt this abuse of terminology for Trunk(Group) and Trunk(Ab). Further information on trunks may be found in the papers =-=[15]-=- and [16] by Fenn, Rourke, and Sanderson. 1.1 Extensions and expansions We begin by devising a suitable general theory of extensions for racks. Rack extensions have been studied before, in particular ... |

8 |
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Citation Context ...odules. Then there is a natural short exact sequence 0 ! M p+q=n H p (C) X H q (D) ! Hn (C X D) ! M p+q=n 1 Tor X 1 (H p (C); H q (D)) ! 0 Proof This proof is a slightly modied version of the one in [=-=20-=-]. Without loss of generality, we may assume that C issat, since there is a natural isomorphism C X D = D X C CHAPTER 5. SEQUENCES 111 given by c d 7! ( 1) pq d c for c 2 (C p ) x ; d 2 (D q ) x ; x ... |

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Citation Context ... in table 7.3 which, when summed, give values of 8 + 8t and 8 + 4s 2 + 4s 3 for the state sum polynomials, thus conrming that CHAPTER 7. APPLICATIONS 123 a b f e c d Figure 7.5: The link 6 2 3 [0] = [=-=-=-2] [1] = [3] 1 1 1 2 t s 2 3 1 1 4 t s 3 5 t s 3 6 1 1 7 t 1 8 1 s 2 [0] = [2] [1] = [3] 9 1 1 10 t s 3 11 1 1 12 t s 2 13 t 1 14 1 s 2 15 t s 3 16 1 1 Table 7.3: Products in ( 6 2 3 ) 6 2 3 and 6 2... |

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Citation Context ...;x + x;x (g) = g; that is x;x = 0 andsx;x (g) + x;x (g) = g for all g 2 A x , and x 2 X . 2 CHAPTER 1. EXTENSIONS 25 Corollary 1.8 Carter, Saito and Kamada's theory of abelian quandle extensions [12] is the special case of the above denition of abelian quandle extensions where the coe cient groups A = A x are the same, the maps x;y are the identity on A, and the mapssy;x are the zero endomorph... |

7 | Cyclic Bordism and Rack Spaces - Flower - 1995 |

1 |
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Citation Context ...There is a well-dened notion of homology and cohomology groups of a rack or quandle, dened in terms of an appropriate classifying space (the `rack space'); these have already been studied [18] [17] [1=-=6]-=-, as has a related theory, that of `quandle (co)homology' [10]. My original objective was to generalise these existing theories and show how they could be regarded as derived functors constructed on r... |

1 |
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Citation Context ... R(B) to get the diagram B UF (B) - - " B R(B) - - B s B @ @ @ @R UFR(B) ? " RB 0 - CHAPTER 3. RESOLUTIONS 75 in which the map s B = " RBB is a natural transformation UF ! UFR such that=-= Lemma 3.20 ([22] Lemma I-=-X.7.1) The morphisms " B and s B induce, for every object A in B a left exact sequence of abelian groups: 0 ! HomA (FR(M);A) s B ! HomB (UF (B); U(A)) " B ! HomB (B; U(A)) Now, each object... |

1 |
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Citation Context ... b 2 X . Crossed modules weresrst studied, in the context of homotopy theory, by Whitehead [29]. More recently, their cohomology has been studied by Carrasco, Cegarra, and Grandjean [7], and by Paoli =-=[24]-=-. It would be interesting to see if a (co)homology theory for augmented racks agrees with crossed-module (co)homology in the appropriate cases. Cotriple homology Another potentially rich topic for fur... |

1 |
An Introduction to Homological Algebra. Number 28
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Citation Context ...that is, every bimorphism (morphism which is both epic and monic) is an isomorphism. Curious readers are directed to any suitable book on category theory or homological algebra (such as [23], [6], or =-=[2-=-8]). 2.2 Beck modules The following result justies our use of the term `rack module' to denote the objects of RModX : Theorem 2.5 The category RModX of rack modules over a (xed) rack X is equivalent t... |