## Computations and homotopical applications of induced crossed modules

Venue: | J. Symb. Comp |

Citations: | 11 - 8 self |

### BibTeX

@ARTICLE{Brown_computationsand,

author = {Ronald Brown and Christopher D Wensley},

title = {Computations and homotopical applications of induced crossed modules},

journal = {J. Symb. Comp},

year = {},

pages = {59--72}

}

### OpenURL

### Abstract

We explain how the computation of induced crossed modules allows the computation of certain homotopy 2-types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications.

### Citations

339 |
Cohomology of Groups, Graduate Texts
- Brown
- 1982
(Show Context)
Citation Context ...n the structure of a crossed module. This fact is of importance in algebraic K-theory. We also note the following fact, shown in various texts on homological algebra or the cohomology of groups, e.g. =-=[6]-=-, and which we relate to topology in section 4: 1.2 A crossed module M = (µ : M → P) determines algebraically a cohomology class kM ∈ H 3 (coker µ, ker µ), called the k-invariant of M, and all element... |

69 |
On the 3-type of a complex
- Lane, Whitehead
- 1956
(Show Context)
Citation Context ...d relative homotopy group functor Π2 : (based pairs of spaces) → (crossed modules) , (X, A, a) ↦→ (∂ : π2(X, A, a) → π1(A, a)) , where a ∈ A ⊆ X (see Section 4). Mac Lane and Whitehead showed in 1950 =-=[22]-=- that crossed modules modelled homotopy 2-types (3-types in their notation) and evidence has grown that crossed modules can be regarded as ‘2-dimensional groups’. Part of this evidence is the 2-dimens... |

67 |
Spaces with finitely many non-trivial homotopy groups
- Loday
- 1982
(Show Context)
Citation Context ...as there is essentially only one category of groups, there are at least five categories of equationally 4defined algebraic structures which are equivalent to crossed modules, namely: • cat 1 -groups =-=[21]-=-; • group-groupoids [11]; • simplicial groups with Moore complex of length 1, [21]; • reduced simplicial T-complexes of rank 2, [15, 3, 24]; • reduced double groupoids with connection [12]. These cate... |

54 | On the connection between the second relative homotopy groups of some related space
- Brown, Higgins
- 1978
(Show Context)
Citation Context ...and evidence has grown that crossed modules can be regarded as ‘2-dimensional groups’. Part of this evidence is the 2-dimensional version of the Van Kampen Theorem proved by Brown and Higgins in 1978 =-=[8]-=-, which allows new computations of homotopy 2-types and so second homotopy groups. This result should be seen as a higher dimensional, non commutative, local-toglobal theorem, illustrating themes in A... |

45 | The classifying space of a crossed complex
- Brown, Higgins
- 1991
(Show Context)
Citation Context ...m in this dimension, on putting A = ΓV , and f : V → ΓV the inclusion. We will apply this Theorem 4.1 to the classifying space of a crossed module, as defined by Loday in [21] or Brown and Higgins in =-=[9]-=-. This classifying space is a functor B assigning to a crossed module M = (µ : M → P) a based CW-space BM with the following properties: 4.2 The homotopy groups of the classifying space of the crossed... |

39 |
On adding relations to homotopy groups
- Whitehead
- 1946
(Show Context)
Citation Context ... a a a a α −1 2 α1 α2 ∼ ❅ β2 ❅■ ❅ a a a a α1 � �✒�� β2 a a ✛ β2 ✲ β1 ✲ β2 ✛ β2 ✲ β1 ✲ β2 ✛ β2 ✲ β1 α −1 2 α1α2 α ∂α2 1 Figure 1: Verification of CM2) for Π2(X, A, a) . ✲ β2 Whitehead’s main result in =-=[27, 28, 29]-=- was: Theorem 1.1 (Whitehead) If X is obtained from A by attaching 2-cells, then π2(X, A, x) is isomorphic to the free crossed π1(A, x)-module on the attaching maps of the 2-cells. Later Quillen obser... |

32 |
Identities among relations
- Brown, Huebschmann
- 1982
(Show Context)
Citation Context ...M, written (m, p) ↦→ m p , satisfying for all m, n ∈ M, p ∈ P the axioms: CM1) µ(m p ) = p −1 (µm)p , CM2) n µm = m −1 nm . When CM1) is satisfied, but not CM2), the structure is a pre-crossed module =-=[10, 19]-=-, having a Peiffer subgroup C generated by Peiffer commutators 〈m, n〉 = m −1 n −1 m n µm , and an associated crossed module (µ ′ : M/C → P) with µ ′ induced by µ. Some standard algebraic examples of c... |

28 |
Théorie de Schreier Superieure
- Breen
- 1992
(Show Context)
Citation Context ...tegory of groupoids. This notion has a long history: the result that crossed modules are equivalent to group-groupoids goes back to Verdier, seems first to have been published in [11], and is used in =-=[5]-=-. The holomorph Aut(M) ⋉ M of a group M is the source of the cat 1 -group associated to the automorphism crossed module (χ : M → Aut(M)). Now a colimit of cat 1 -groups colimi(ei; ti, hi : Gi → Ri) is... |

27 |
Two-dimensional homotopy and combinatorial group theory, volume 197
- Hog-Angeloni, Metzler, et al.
- 1993
(Show Context)
Citation Context ...M, written (m, p) ↦→ m p , satisfying for all m, n ∈ M, p ∈ P the axioms: CM1) µ(m p ) = p −1 (µm)p , CM2) n µm = m −1 nm . When CM1) is satisfied, but not CM2), the structure is a pre-crossed module =-=[10, 19]-=-, having a Peiffer subgroup C generated by Peiffer commutators 〈m, n〉 = m −1 n −1 m n µm , and an associated crossed module (µ ′ : M/C → P) with µ ′ induced by µ. Some standard algebraic examples of c... |

26 |
crossed modules and the fundamental groupoid of a topological group
- Brown, Spencer, et al.
- 1976
(Show Context)
Citation Context ...t 1 -groups [21]; • group-groupoids [11]; • simplicial groups with Moore complex of length 1, [21]; • reduced simplicial T-complexes of rank 2, [15, 3, 24]; • reduced double groupoids with connection =-=[12]-=-. These categories have various geometric models. The 2-cells of some of these are illustrated in the following pictures: e 0 ∪ e 1 ∪ e 2 e 0 ± ∪ e1 ± ∪ e2 2-simplex square ✬✩ � ✻ ✫✪ ✬✩ ✲ � ✫✪ ✲ � � ✁... |

25 | On induced crossed modules, and the homotopy 2-type of mapping cones
- Brown, Wensley
- 1995
(Show Context)
Citation Context ...ssed modules, the discussion of which is the theme of this paper. Our main emphasis in this paper is on induced crossed modules, which were defined in [8] and studied further in papers by the authors =-=[13, 14]-=-. Given the crossed module M = (µ : M → P) and a morphism of groups ι : P → Q, the induced crossed module ι∗M has the form (∂ : ι∗M → Q), a crossed module over Q, and comes with a morphism of crossed ... |

18 | Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types
- Brown, Wensley
- 1996
(Show Context)
Citation Context ...ssed modules, the discussion of which is the theme of this paper. Our main emphasis in this paper is on induced crossed modules, which were defined in [8] and studied further in papers by the authors =-=[13, 14]-=-. Given the crossed module M = (µ : M → P) and a morphism of groups ι : P → Q, the induced crossed module ι∗M has the form (∂ : ι∗M → Q), a crossed module over Q, and comes with a morphism of crossed ... |

17 | Double groupoids and crossed modules - Brown, Spencer - 1976 |

14 |
2001, Graphs of Groups: Word Computations and Free Crossed Resolutions
- Moore
(Show Context)
Citation Context ...4, with XMod2 included with the 4.3 2release. Related libraries include Heyworth’s IdRel [17] for computing identities among the relators of a finitely presented group, and Moore’s GpdGraph and XRes =-=[23]-=- for computing with finite groupoids; group and groupoid graphs; and crossed resolutions. These libraries are available at the HDDA website [18]. 1 Crossed modules A crossed module M (over P) consists... |

11 | G-groupoids, crossed modules and the fundamental groupoid of a topological group - Brown, Spencer - 1976 |

11 |
Note on a previous paper entitled `On adding relations to homotopy groups
- Whitehead
- 1946
(Show Context)
Citation Context ...= (I × {0, 1}) ∪ ({1} ×I) ⊂ I 2 . Each such α is a map from the unit square I 2 to the space X mapping three sides of the square to the point a and the fourth side to a loop at a. Whitehead showed in =-=[28]-=- that there is a crossed module Π2(X, A, a) with boundary map ∂ : π2(X, A, a) → π1(A, a), α ↦→ β = α(I × {0}) . The image of α1 ∈ π2(X, A, a) under the action of β2 ∈ π1(A, a) is illustrated in the ri... |

10 |
Double groupoids and crossed modules’, Cah
- Brown
- 1976
(Show Context)
Citation Context ...only one category of groups, there are at least five categories of equationally 4defined algebraic structures which are equivalent to crossed modules, namely: • cat 1 -groups [21]; • group-groupoids =-=[11]-=-; • simplicial groups with Moore complex of length 1, [21]; • reduced simplicial T-complexes of rank 2, [15, 3, 24]; • reduced double groupoids with connection [12]. These categories have various geom... |

9 |
et al, Gap: groups, Algorithms
- Schönert
- 1993
(Show Context)
Citation Context ... of this project saw the production of the library XMod1, containing functions for crossed modules and their derivations and for cat 1 -groups and their sections. The manual for XMod1 was included in =-=[26]-=- as Chapter 73. In particular, Alp [1] enumerated all isomorphism classes of cat 1 -structures on groups of order at most 47. This library has recently been rewritten for GAP4, with XMod2 included wit... |

8 |
Enumeration of Cat1 -groups of low order
- Alp, Wensley
(Show Context)
Citation Context ...t ′ , h ′ . When C = (e; t, h : G → R) and ι : R → Q is an inclusion, the induced cat 1 -group ι∗C is obtained as the pushout of cat 1 -morphisms (e, idR) : CR → C and (ι, ι) : CR → CQ . See Alp [1], =-=[2]-=- for further details. Further investigation is needed to see whether the use of cat 1 -groups can be shown to be more efficient than the direct method for the computation of some colimits of crossed m... |

8 | Computability and Complexity - Jones - 1997 |

7 |
Identities among relations
- Pride
- 1991
(Show Context)
Citation Context ... A considerable amount of work has been done on this case, because of the connections with identities among relations, and methods such as transversality theory and “pictures” have proved successful (=-=[10, 25]-=-), particularly in the homotopy theory of 2-dimensional complexes [19]. However, the only route so far available to the wider geometric applications of induced crossed modules is Theorem 4.1. We also ... |

6 |
Logged rewriting and identities among relators. In
- Heyworth, Wensley
- 2003
(Show Context)
Citation Context ...m classes of cat 1 -structures on groups of order at most 47. This library has recently been rewritten for GAP4, with XMod2 included with the 4.3 2release. Related libraries include Heyworth’s IdRel =-=[17]-=- for computing identities among the relators of a finitely presented group, and Moore’s GpdGraph and XRes [23] for computing with finite groupoids; group and groupoid graphs; and crossed resolutions. ... |

5 |
crossed modules, cat1-groups: applications of computational group theory
- GAP
- 1997
(Show Context)
Citation Context ...the library XMod1, containing functions for crossed modules and their derivations and for cat 1 -groups and their sections. The manual for XMod1 was included in [26] as Chapter 73. In particular, Alp =-=[1]-=- enumerated all isomorphism classes of cat 1 -structures on groups of order at most 47. This library has recently been rewritten for GAP4, with XMod2 included with the 4.3 2release. Related libraries... |

5 | Homotopy theory, and change of base for groupoids and multiple groupoids’, Applied categorical structures
- Brown
- 1996
(Show Context)
Citation Context ...s over P) . This functor has a left adjoint ι∗ : (crossed modules over P) → (crossed modules over Q) , which gives our induced crossed module. This construction can be described as a “change of base” =-=[7]-=-. To compute a colimit colimi(µi : Mi → Pi), one forms the group P = colimiPi, and uses the canonical morphisms φi : Pi → P to form the family of induced crossed Pmodules ((µi)∗ : (φi)∗Mi → P). The co... |

4 | Cohomology of Groups volume 87 - Brown - 1982 |

3 | Mathematics in the 20th Century - Atiyah |

2 |
Multiple compositions for higher dimensional groupoids
- Dakin
- 1977
(Show Context)
Citation Context ...es which are equivalent to crossed modules, namely: • cat 1 -groups [21]; • group-groupoids [11]; • simplicial groups with Moore complex of length 1, [21]; • reduced simplicial T-complexes of rank 2, =-=[15, 3, 24]-=-; • reduced double groupoids with connection [12]. These categories have various geometric models. The 2-cells of some of these are illustrated in the following pictures: e 0 ∪ e 1 ∪ e 2 e 0 ± ∪ e1 ± ... |

2 | GAP—Groups, Algorithms - Schönert - 1995 |

2 |
Simplicial T-Complexes
- Ashley
- 1976
(Show Context)
Citation Context ...es which are equivalent to crossed modules, namely: • cat 1 -groups [21]; • group-groupoids [11]; • simplicial groups with Moore complex of length 1, [21]; • reduced simplicial T-complexes of rank 2, =-=[15, 3, 24]-=-; • reduced double groupoids with connection [12]. These categories have various geometric models. The 2-cells of some of these are illustrated in the following pictures: e 0 ∪ e 1 ∪ e 2 e 0 ± ∪ e1 ± ... |

1 | Simplicial T -complexes - Ashley - 1978 |

1 | Induced Crossed Modules 13 - Heyworth, Wensley |

1 |
A Dold-Kan theorem for crossed complexes
- Tie, G
- 1989
(Show Context)
Citation Context ...es which are equivalent to crossed modules, namely: • cat 1 -groups [21]; • group-groupoids [11]; • simplicial groups with Moore complex of length 1, [21]; • reduced simplicial T-complexes of rank 2, =-=[15, 3, 24]-=-; • reduced double groupoids with connection [12]. These categories have various geometric models. The 2-cells of some of these are illustrated in the following pictures: e 0 ∪ e 1 ∪ e 2 e 0 ± ∪ e1 ± ... |

1 |
Mathematics in the 20th
- Atiyah
- 2002
(Show Context)
Citation Context ...ew computations of homotopy 2-types and so second homotopy groups. This result should be seen as a higher dimensional, non commutative, local-toglobal theorem, illustrating themes in Atiyah’s article =-=[4]-=-. It is interesting to note that the computation of these second homotopy groups is obtained through the computation of a larger non commutative structure. This work also throws emphasis on the proble... |