## N-FOLD ČECH DERIVED FUNCTORS AND GENERALISED HOPF TYPE FORMULAS

Citations: | 4 - 0 self |

### BibTeX

@MISC{Donadze_n-foldčech,

author = {Guram Donadze and Nick Inassaridze and Timothy Porter},

title = {N-FOLD ČECH DERIVED FUNCTORS AND GENERALISED HOPF TYPE FORMULAS},

year = {}

}

### OpenURL

### Abstract

Abstract. In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Čech derived functors of the lower central series functors Zk. The paper ends with an application to algebraic K-theory. Introduction and Summary The well known Hopf formula for the second integral homology of a group says that for a given group G there is an isomorphism H2(G) ∼ = R ∩ [F, F]

### Citations

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Citation Context ...the crossed n-cube of groups M, (Proposition 12). Section 4 is devoted to the investigation of some properties of the mapping cone complex of a morphism of (non-abelian) group complexes introduced in =-=[17]-=-. In particular, for a given morphism of pseudosimplicial groups α : G∗ → H∗ the natural morphism κ : NM∗(α) → C∗(˜α) induces isomorphisms of their homology groups, where C∗(˜α) is the mapping cone co... |

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Citation Context ...htforward from Theorem 9. n−1 ∩ Ker dn−1 i ∩ Γk(Fn−1) i=0 Dk(Fn−1; Ker d n−1 0 , . . . , Ker d n−1 n−1) , n ≥ 1 . □ □ 3. Crossed n-cubes of groups The following definition is due to Ellis and Steiner =-=[11]-=- (see also [25]). A crossed ncube of groups is a family {MA : A ⊆ 〈n〉} of groups, together with homomorphisms, µi : MA → MA\{i} for i ∈ 〈n〉, A ⊆ 〈n〉 and functions h : MA × MB −→ MA∪B for A, B ⊆ 〈n〉, s... |

31 |
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Citation Context ...ossed module of crossed modules or a morphism of complexes of length 1 satisfying the conditions (∗). The construction above gives a complex C∗(M) of length 2. (This has a 2-crossed module structure, =-=[6]-=-, as noted by Conduché, see also [21].) Proceeding by induction, suppose for any crossed (n−1)-cube M we have constructed a complex C∗(M) of length n − 1. Now let M be a crossed n-cube and consider it... |

26 |
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Citation Context ...n [22] (see also [12] and, here, our Section 1) as an algebraic analogue of the Čech (co)homology construction of open covers of topological spaces with coefficients in sheaves of abelian groups (see =-=[29]-=-). It is well known that the Čech cohomlogy of topological spaces with coefficients in sheaves is closely related to sheaf cohomology of topological spaces, in particular this relation is expressed by... |

23 |
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Citation Context ...Theorem 9. n−1 ∩ Ker dn−1 i ∩ Γk(Fn−1) i=0 Dk(Fn−1; Ker d n−1 0 , . . . , Ker d n−1 n−1) , n ≥ 1 . □ □ 3. Crossed n-cubes of groups The following definition is due to Ellis and Steiner [11] (see also =-=[25]-=-). A crossed ncube of groups is a family {MA : A ⊆ 〈n〉} of groups, together with homomorphisms, µi : MA → MA\{i} for i ∈ 〈n〉, A ⊆ 〈n〉 and functions h : MA × MB −→ MA∪B for A, B ⊆ 〈n〉, such that if a b... |

18 | Homotopical excision and the Hurewicz theorem for n-cubes of spaces
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Citation Context ... 30], but perhaps the most successful one, giving formulas in all dimensions, was by Brown and Ellis, [3]. They used topological methods, and in particular the Hurewicz theorem for n-cubes of spaces, =-=[5]-=-, which itself is an application of the generalised van Kampen theorem for diagrams of spaces [4]. The end result was: 1991 Mathematics Subject Classification. 18G50, 18G10. Key words and phrases. Čec... |

13 |
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Citation Context ...rdinary Čech complex construction Č to the morphism of Čech complexes Č(F{1} )∗ → Č(F{1} )∗. Now applying the right exact functor Zk dimension-wise, denote the resulted bisimplicial group Zk(F∗∗). By =-=[27]-=- there is a spectral sequence E 2 pq =⇒ L2−fold p+q Zk(G) , where E2 0q = 0, q > 0 and E2 p0 ∼ = L1−fold p Zk(G), p ≥ 0. Hence there is an isomorphism L 2−fold 1 Zk(G) ∼ = → L 1−fold 1 Zk(G). Now the ... |

10 | Hopf formulae for the higher homology of a group
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(Show Context)
Citation Context ...ge Timothy Porter Mathematics Division, School of Informatics, University of Wales Bangor, Bangor, Gwynedd LL57 1UT, United Kingdom. t.porter@bangor.ac.uk Abstract. In 1988, Brown and Ellis published =-=[3]-=- a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that con... |

10 | Applications of Peiffer pairing in the Moore complex of a simplicial group
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Citation Context ...derived functors, proved the same result, [10]. The similarity between this formula and the formulae given by Mutlu and the third author for the various homotopy invariants of a simplicial group (see =-=[19, 20]-=-) suggested that there should be a purely algebraic proof of this, which hopefully would generalise further. Examining the classical case (n = 1), and the proof of the usual Hopf formula, showed a lin... |

9 |
Non-abelian homological algebra and its applications
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(Show Context)
Citation Context ...ich hopefully would generalise further. Examining the classical case (n = 1), and the proof of the usual Hopf formula, showed a link with the Čech derived functors of the abelianisation functor, (cf. =-=[12]-=-). Trying to derive this result purely algebraically and to obtain Hopf type formulas for some more general situations, we suspected that the conditions given above for Theorem BE were not sufficient ... |

6 |
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Citation Context ...tors on the category of groups and generalise the abelianization functor, so their non-abelian left derived functors, LnZk, n ≥ 0, generalise the group homology functors Hn, n ≥ 1, cf., for instance, =-=[1]-=-. In [13], a Hopf-like formula is proved for the second Conduché-Ellis homology of precrossed modules using Čech derived functors. The main goal of this paper is to develop this method further, and by... |

3 |
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Citation Context ...i∈A Hn+1(G) ∼ = 1≤i≤n n ∩ i=1 Ri ∩ [F, F ] ∏ [ ∩ Ri, ∩ Ri] i∈A i/∈A . A⊆〈n〉 Later, Ellis using mainly algebraic means, and, in particular, his hyper-relative derived functors, proved the same result, =-=[10]-=-. The similarity between this formula and the formulae given by Mutlu and the third author for the various homotopy invariants of a simplicial group (see [19, 20]) suggested that there should be a pur... |

3 |
Crossed squares and 2-crossed modules
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(Show Context)
Citation Context ... morphism of complexes of length 1 satisfying the conditions (∗). The construction above gives a complex C∗(M) of length 2. (This has a 2-crossed module structure, [6], as noted by Conduché, see also =-=[21]-=-.) Proceeding by induction, suppose for any crossed (n−1)-cube M we have constructed a complex C∗(M) of length n − 1. Now let M be a crossed n-cube and consider it as a crossed module of crossed (n − ... |

3 |
A generalized Hopf formula for higher homology groups
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(Show Context)
Citation Context ...2(G) ∼ = R ∩ [F, F ] [F, R] where R ↣ F ↠ G is a free presentation of the group G. Several alternative generalisations of this classical Hopf formula to higher dimensions were made in various papers, =-=[9, 28, 30]-=-, but perhaps the most successful one, giving formulas in all dimensions, was by Brown and Ellis, [3]. They used topological methods, and in particular the Hurewicz theorem for n-cubes of spaces, [5],... |

2 |
Crossed n-fold extensions of groups, n-fold extensions of modules, and higher multipliers
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(Show Context)
Citation Context ...2(G) ∼ = R ∩ [F, F ] [F, R] where R ↣ F ↠ G is a free presentation of the group G. Several alternative generalisations of this classical Hopf formula to higher dimensions were made in various papers, =-=[9, 28, 30]-=-, but perhaps the most successful one, giving formulas in all dimensions, was by Brown and Ellis, [3]. They used topological methods, and in particular the Hurewicz theorem for n-cubes of spaces, [5],... |

2 |
Spectral sequence for epimorphism
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(Show Context)
Citation Context ...cohomology of topological spaces, in particular this relation is expressed by spectral sequences [29]. Some applications of Čech derived functors to group (co)homology theory and Ktheory are given in =-=[22, 23, 24]-=-. In this section we generalise the notion of the Čech derived functors to that of the n-fold Čech derived functors of an endofunctor on the category of groups. Based on this notion we get a new purel... |

2 | Simplicial degree of functors
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(Show Context)
Citation Context ...it using the Čech derived functors. Theorem 3. Let G be a group and R i ↣ F ↠ G be a free presentation of the group G. Then there is an isomorphism of groups H2(G) ∼ = R ∩ [F, F ] [F, R] Proof. Using =-=[22, 24]-=- (see also Theorem 2.39(ii), [12]), there is an isomorphism H2(G) ∼ = L1Ab(G) , where L1Ab is the first Čech derived functor of the group abelianization functor. Now Lemma 1 and Proposition 2 implies ... |

2 |
Presentaciones libres y H2n(G), Publ
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- 1986
(Show Context)
Citation Context ...2(G) ∼ = R ∩ [F, F ] [F, R] where R ↣ F ↠ G is a free presentation of the group G. Several alternative generalisations of this classical Hopf formula to higher dimensions were made in various papers, =-=[9, 28, 30]-=-, but perhaps the most successful one, giving formulas in all dimensions, was by Brown and Ellis, [3]. They used topological methods, and in particular the Hurewicz theorem for n-cubes of spaces, [5],... |

1 |
The Central Series for Peiffer Commutators
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- 1990
(Show Context)
Citation Context ...iven group G the (lower) central series Γk = Γk(G), G = Γ1 ⊇ Γ2 ⊇ · · · ⊇ Γk ⊇ · · · of G is defined inductively by Γk = ∏ [Γi, Γj]. The well-known Witt-Hall identities i+j=k on commutators (see e.g. =-=[2]-=-) imply that Γk = [G, Γk−1]. Let Gr denote the category of groups. Let us define higher abelianization type functors Zk : Gr → Gr, k ≥ 2 by Zk(G) = G/Γk(G) for any G ∈ ob Gr and where Zk(α) is the nat... |

1 |
Simplicial crossed modules and mapping cones, preprint, Université de Rennes I
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(Show Context)
Citation Context ...duced morphism of the Moore complexes and NM∗(α) is the Moore complex of a new pseudosimplicial group constructed using α (Proposition 13). (Here similar results have recently been found by Conduché, =-=[8]-=-.) Using this result we derive purely algebraicly the result of [17], (3.4. Proposition), giving for a crossed n-cube of groups M an isomorphism between the homotopy groups of E (n)(M)∗ and the corres... |

1 | More about homological properties of precrossed modules - Inassaridze, Khmaladze |

1 |
Derived functors and algebraic
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(Show Context)
Citation Context ...efine the functor Z∞ : Gr → Gr as follows: for a given group G, Z∞(G) = limZj(G); for a given group homomorphism λ : G → H, Z∞(λ) is the ←− j group homomorphism induced by the Zj(λ). It is known from =-=[14]-=- (see also [12]) that the non-abelian left derived functors L P ∗ Z∞ of the functor Z∞ : Gr → Gr are isomorphic to Quillen’s K-functors. Thus using Theorem 24 we deduce that there is a short exact seq... |

1 | Multirelative K-theory and axioms for the K-theory of rings, K-theory Preprint Archives 137
- Keune
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(Show Context)
Citation Context ... groups (F ; R1, . . . , Rn) satisfies the following condition: F � ∏ Ri ∼ = FA for all A ⊆ 〈n〉 , i∈A i.e. the n-cube of groups F is induced by the normal (n+1)-ad of groups (F ; R1, . . . , Rn) (see =-=[15]-=-). Now let F be an n-presentation of the group G. Applying Č (see Section 1) in the n-independent directions, this construction leads naturally to an augmented nsimplicial group. Taking the diagonal o... |

1 |
A.R.-Grandjeán, Crossed modules and homology, J.Pure Appl. Algebra 95
- Ladra
- 1994
(Show Context)
Citation Context ...1, . . . , rn, f) = (r1 · · · rnf, r2 · · · rnf, . . . , rnf, f) for all n ≥ 1 and (r1, . . . , rn, f) ∈ E(R ↩→ F )n. It is easy to check that λ∗ is isomorphism of simplicial groups. □ We recall from =-=[16]-=- that a crossed module µ : M → P is called abelian if P is an abelian group and the action of P on M is trivial. This implies that M is also abelian. Let us denote the category of abelian crossed modu... |

1 |
On non-abelian derived functors, Proc
- Pirashvili
- 1979
(Show Context)
Citation Context ...it using the Čech derived functors. Theorem 3. Let G be a group and R i ↣ F ↠ G be a free presentation of the group G. Then there is an isomorphism of groups H2(G) ∼ = R ∩ [F, F ] [F, R] Proof. Using =-=[22, 24]-=- (see also Theorem 2.39(ii), [12]), there is an isomorphism H2(G) ∼ = L1Ab(G) , where L1Ab is the first Čech derived functor of the group abelianization functor. Now Lemma 1 and Proposition 2 implies ... |