## Homotopical excision, and Hurewicz theorems, for n-cubes of spaces (1987)

Venue: | Proc. London Math. Soc |

Citations: | 22 - 9 self |

### BibTeX

@ARTICLE{Brown87homotopicalexcision,,

author = {Ronald Brown and Jean-louis Loday},

title = {Homotopical excision, and Hurewicz theorems, for n-cubes of spaces},

journal = {Proc. London Math. Soc},

year = {1987},

pages = {5--4}

}

### Years of Citing Articles

### OpenURL

### Abstract

The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into an exact sequence.

### Citations

78 | Colimit theorems for relative homotopy groups
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...ps πp+1(A, A∩B), πq+1(B, A∩B) may be non-abelian, and acts on the other group. In the description of the critical group, the usual tensor product must be replaced by the tensor product G⊗H defined in =-=[5, 6]-=-, which involves actions of G on H and H on G. This description of πp+q+1(X; A, B) is a special case of a description of the hyper-relative group πn+1(X; A1, . . . , An) of a ‘connected’ excisive (n +... |

56 | On the connection between the second relative homotopy group some related space
- Brown, Higgins
- 1978
(Show Context)
Citation Context ... that of changing base rings in module theory, and also to that of inducing representations in group representation theory. In order to explain the intuitive basis of the construction, we recall from =-=[4]-=- the notion of induced crossed module. Let Q be a group, and let XMQ be the category of crossed Q-modules. Let f : P → Q be a homomorphism of groups. Then pullback (fibre product) defines a functor f∗... |

51 |
Kampen theorems for diagrams of spaces, Topology 26
- Brown, Loday, et al.
- 1987
(Show Context)
Citation Context ...X ∪ {e2 λ }, X) as a free crossed π1X-module on the characteristic maps of the 2-cells e2 λ . All of our homotopical results are deductions from the Van Kampen theorem for n-cubes of spaces proved in =-=[7]-=-. This theorem states that the fundamental catn-group functor Π from ncubes of spaces to catn-groups preserves certain colimits. The main trick for the applications is to choose the colimit of n-cubes... |

37 |
Higher-dimensional crossed modules and the homotopy groups of (n+ 1)-ads
- Ellis, Steiner
- 1987
(Show Context)
Citation Context ...commutators respectively. It is a theorem of D. Guin-Waléry and J.-L. Loday that this association gives an equivalence between cat2-groups and crossed squares [10, §5]. Crossed n-cubes are defined in =-=[8,9]-=-, and an equivalence between them and catn-groups is proved there. We shall use some information on crossed 3-cubes in §4. In [10, Theorem 1.4] there is defined a functor from n-cubes of fibrations to... |

15 | Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups’, Topology 23
- BROWN
- 1984
(Show Context)
Citation Context ...ase the equivalence between cat 2 -groups and crossed squares given in [9] (see §2 above). To this end we first recall that the category XMQ of crossed Q-modules has a coproduct which is described in =-=[3]-=- as follows. Let M, N be crossed Q-modules. Then N acts on M, and M acts on N, via the given actions of Q. Let M⋊N denote the semidirect product with injections i ′ : M → M⋊N, m ↦→ (m, 1), and j ′ : N... |

14 |
Spaces with finitely many homotopy groups
- Loday
- 1984
(Show Context)
Citation Context ... bibj = bjbi, if i ̸= j. The group G is called the big group of the catn-group G and will be written B(G); the intersection L(G) = ⋂n i=1 Ker si is also important in applications and is emphasized in =-=[10, 7]-=-. A morphism f : G → H of catn-groups is a homomorphism f : G → H of groups which commutes with the si, bi, for i = 1, . . . , n. So we have a category (catn-groups) of catn -groups. For explicit calc... |

11 |
Crossed modules and their higher dimensional analogues
- Ellis
- 1984
(Show Context)
Citation Context ..., as required. The case where p = 1, q > 1 is similar. ✷ We now give an application combining Theorem 4.1 with the case n = 3 of the equivalence between catn-groups and crossed n-cubes established in =-=[8, 9]-=-. We do not give definitions here since we need only one example. Example 4.4 [8,9]. Consider a crossed 3-cube of the form � 0 L ��� � G G ��� � G �� �� G ��� � G G ��� � � G�� �� �� �� �� � �� �� ��... |

8 |
Excision homotopique en basse dimension
- Brown, Loday
- 1984
(Show Context)
Citation Context ...ps πp+1(A, A∩B), πq+1(B, A∩B) may be non-abelian, and acts on the other group. In the description of the critical group, the usual tensor product must be replaced by the tensor product G⊗H defined in =-=[5, 6]-=-, which involves actions of G on H and H on G. This description of πp+q+1(X; A, B) is a special case of a description of the hyper-relative group πn+1(X; A1, . . . , An) of a ‘connected’ excisive (n +... |

4 |
ALGEBRAIC TOPOLOGY - A STUDENT’S GUIDE
- Adams
- 1972
(Show Context)
Citation Context ...�� �� �� HOMOTOPICAL EXCISION, AND HUREWICZ THEOREMS 179 n-comer Y, if it exists, is called the pushout of Y. Then Y and the universal natural transformation Y → colim Y define an n-cube X in C with X=-=(1)-=- = colim Y and for α ̸= 1, X(α) = Y(α). Such an X we call an n-pushout, or pushout n-cube, in C. Note that the diagrams for 2-corners and 3-corners are of the following forms respectively: �� �� � ���... |

3 | The homotopy groups of a triad III - Blakers, Massey - 1953 |