## Colimit Theorems for Relative Homotopy Groups (2008)

Citations: | 75 - 34 self |

### BibTeX

@MISC{Brown08colimittheorems,

author = {Ronald Brown and Philip J. Higgins},

title = {Colimit Theorems for Relative Homotopy Groups},

year = {2008}

}

### OpenURL

### Abstract

This is the second of two papers whose main purpose is to prove a generalisation to all dimensions of the Seifert-Van Kampen theorem on the fundamental group of a union of spaces. The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω-groupoids and C of crossed complexes were defined, and the principal result

### Citations

122 | On the algebra of cubes
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...ion This is the second of two papers whose main purpose is to prove a generalisation to all dimensions of the Seifert-Van Kampen theorem on the fundamental group of a union of spaces. The first paper =-=[10]-=- (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω-groupoids and C of crossed complexes were defined, and the principal result of [10] was an equivale... |

105 |
Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete
- Gabriel, Zisman
- 1967
(Show Context)
Citation Context ...1 can be recovered as induced by the maps δλ : Cλ n → Cλ n−1 , for all λ. Colimits of groupoids are easily described by generators and relations and are as readily computed as colimits of groups (see =-=[15, 16, 17]-=-). Colimits in U and C, U are less transparent and we analyse their structure further by the use of induced modules and induced crossed modules (over groupoids). Given a module (M, H) and a morphism o... |

52 |
Combinatorial homotopy (I
- Whitehead
- 1949
(Show Context)
Citation Context ...f thin elements in ϱX∗. For the applications, this colimit theorem is recast in terms of the closely related invariant ΠX∗, the fundamental crossed complex of X∗ (studied under other names in [3] and =-=[23]-=-). We show in Section 5 that γϱX∗ is naturally isomorphic to ΠX∗, and hence obtain colimit theorems for ΠX∗ (Theorems C and D of Section 5). In the proofs of all these results, one of the key ingredie... |

50 | On the connection between the second relative homotopy groups of some related spaces
- Brown, Higgins
- 1978
(Show Context)
Citation Context ...oupoids The groupoid version of the Van Kampen theorem [4, 8.4.2], gives useful results for nonconnected spaces, but still requires a ‘representativity’ condition in dimension 0. The union theorem of =-=[7]-=-, 10�� � which computes second relative homotopy groups, requires conditions in dimension 0 and 1. It is thus not surprising that our general union theorem requires conditions in all dimensions. A fi... |

43 | Tensor products and homotopies for ω-groupoids and crossed complexes
- Brown, Higgins
- 1987
(Show Context)
Citation Context ...cal tool in later proofs. 1 The full account of such a notion in which a homotopy ft of filtered maps f0, f1 should satisfy ft(Xn) ⊆ Yn+1, in analogy with cellular homotopies, was given in references =-=[26,27]-=-, and also in [25]. 7First let C be an r-cell in the n-cube I n . Two (r − 1)-faces of C are called opposite if they do not meet. A partial box in C is a subcomplex B of C generated by one (r − 1)-fa... |

37 | On adding relations to homotopy groups - Whitehead - 1941 |

24 | Topology and Groupoids
- Brown
- 2006
(Show Context)
Citation Context ...ce (∗) may be extended to the right by δ r n : πn(Xr, Xn, ν) → πn−1(Xn, Xn−1, ν). So ir n surjective implies δr n (this is still true for n = 2 since we have the rule δr 2a = δr 2 ab−1 ∈ Im jr 2 (see =-=[5]-=-). Hence the composite δ 3 2δ 4 3 . . . δ n n−1 : πn(Xr, Xn, ν) → π1(X2, X1, ν) b if and only if is injective. Therefore πn(Xr, Xn, ν) = 0. ✷ 6 Colimits of crossed complexes The usefulness of Theorems... |

16 |
A certain exact sequence
- WHITEHEAD
- 1950
(Show Context)
Citation Context ...rewicz theorem; in Section 8 we give a proof of the absolute theorem in the present context, and relate the homotopy exact sequence of the fibration p : RX∗ → ϱX∗ to work of Blakers [3] and Whitehead =-=[22, 24]-=-. In Section 9 we establish that an ω-groupoid is isomorphic to some ϱX∗, and that any crossed complex is isomorphic to some ΠY∗; hence these constructions generalise constructions of Eilenberg-Mac La... |

15 |
complexes and multiple groupoid structures
- Dakin, Kan
- 1997
(Show Context)
Citation Context ...lises a functor of Dold-Kan [19, Theorem 22.4] from chain complexes to simplicial abelian groups. Acknowledgement We must again thank Keith Dakin for the insights afforded by his work on T -complexes =-=[12]-=- and for discussions on the problems involved in analysing the structures possessed by ϱX∗. We must thank Nicholas Ashley for discussions which led to considerable improvements in some proofs (in Sect... |

14 |
Some Relations between Homology and Homotopy Groups
- Blakers
- 1948
(Show Context)
Citation Context ...erties of thin elements in ϱX∗. For the applications, this colimit theorem is recast in terms of the closely related invariant ΠX∗, the fundamental crossed complex of X∗ (studied under other names in =-=[3]-=- and [23]). We show in Section 5 that γϱX∗ is naturally isomorphic to ΠX∗, and hence obtain colimit theorems for ΠX∗ (Theorems C and D of Section 5). In the proofs of all these results, one of the key... |

14 |
Presentations of groupoids, with applications to groups
- Higgins
- 1964
(Show Context)
Citation Context ...1 can be recovered as induced by the maps δλ : Cλ n → Cλ n−1 , for all λ. Colimits of groupoids are easily described by generators and relations and are as readily computed as colimits of groups (see =-=[15, 16, 17]-=-). Colimits in U and C, U are less transparent and we analyse their structure further by the use of induced modules and induced crossed modules (over groupoids). Given a module (M, H) and a morphism o... |

11 |
Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids
- Brown, Higgins, et al.
(Show Context)
Citation Context ...by Π for the fundamental crossed complex of a filtered space. We also change the term ‘homotopy full’ to ‘connected’, and the term ‘filtered homotopy’ to ‘thin homotopy’, to agree with terminology in =-=[25]-=- in the additional bibliography. † r.brown@bangor.ac.uk 1on it, in a natural, geometric way, structures of connections Γj and compositions +i, satisfying rules given in [10, Section 1]. In particular... |

10 | On the second relative homotopy group of an adjunction space: an exposition of a Theorem of J.H.C
- Brown
- 1980
(Show Context)
Citation Context ... [21, 23] is still interesting because of its use of the fundamental group of the complement of a link obtained by using methods essentially of transversality; an exposition of this proof is given in =-=[6]-=-. 238 Homotopy and homology There are standard definitions of homology groups for any cubical complex, and of homotopy groups for Kan complexes (cubical complexes satisfying Kan’s extension condition... |

8 | On the van Kampen theorem - Crowell - 1959 |

7 |
Sur les complexes croisés d’homotopie associés a quelques espaces filtrés
- Brown, Higgins
- 1978
(Show Context)
Citation Context ... −→ (crossed complexes) ( ) λ ←− (ω-groupoids) of which the first is given in [2] and the second in [8, 10]. Further it is proved in [2] that Nϱ △ X∗ = πX∗, and we prove in Section 5, as announced in =-=[9]-=-, that λϱX∗ = ΠX∗. Thus Theorem B △ follows from these facts and Theorem B, and at the time of writing no other proof of Theorem B △ is known. 5 The union theorem for crossed complexes In order to int... |

5 | Simplicial T-complexes and crossed complexes - Ashley, D - 1978 |

4 |
and projective crossed modules
- Ratcliffe, Free
- 1980
(Show Context)
Citation Context ..., no other proof of the case n = 2 of Theorem E is known, although a proof of Whitehead’s theorem, that π2(U ∪ {e 2 α}, U) is a free crossed module, has been given by J. Ratcliffe in his Ph.D. thesis =-=[20]-=- using methods of covering spaces, the relative Hurewicz theorem, and a homological characterisation of free crossed modules. Whitehead’s proof [21, 23] is still interesting because of its use of the ... |

3 |
Four applications of the self-obstruction invariant
- Adams
- 1956
(Show Context)
Citation Context ...al filtration. A cellular map of CW -complexes is then a filtered map of the associated filtered spaces. Let In be the standard n-cube with its standard cell structure as a product of n copies of I = =-=[0, 1]-=-. Then the filtered space consisting of In with its skeletal filtration In 0 ⊆ In 1 ⊆ In 2 ⊆ · · · will be written In ∗ . We also write ∂ (In) for the boundary of In , i.e. the subcomplex In n−1 . The... |

3 | Elements of modern topology (McGraw-Hill - Brown - 1968 |

3 | Simplicial Methods in Algebraic Topology - May - 1967 |

3 | Sur les complexes croisés, ω-groupoïdes, et T -complexes - Brown, Higgins - 1977 |

1 |
Lectures in algebraic topology
- Federer
- 1962
(Show Context)
Citation Context ... a cubical complex A is defined in a manner similar to that of the simplicial case [15], using identifications involving only the face operators ∂τ i and degeneracy operators εj. Details are given in =-=[14]-=-, where it is also proved that if X is a space and KX is the singular cubical complex of X, then the natural map jx : |KX| → X induces an isomorphism of homotopy groups. It is proved in [18] that if A... |

1 |
Categories and Groupoids (Van Nostrand
- Higgins
- 1971
(Show Context)
Citation Context ...1 can be recovered as induced by the maps δλ : Cλ n → Cλ n−1 , for all λ. Colimits of groupoids are easily described by generators and relations and are as readily computed as colimits of groups (see =-=[15, 16, 17]-=-). Colimits in U and C, U are less transparent and we analyse their structure further by the use of induced modules and induced crossed modules (over groupoids). Given a module (M, H) and a morphism o... |

1 |
sets and semi-cubical sets
- Hintze, Polysets
- 1973
(Show Context)
Citation Context ... given in [14], where it is also proved that if X is a space and KX is the singular cubical complex of X, then the natural map jx : |KX| → X induces an isomorphism of homotopy groups. It is proved in =-=[18]-=- that if A is a Kan cubical complex, then the natural map iA : A → K|A| induces isomorphisms of homotopy groups. 3 So if (A, B) is a pair of Kan cubical complexes, then the natural map i : (A, B) → (K... |