NFOLD C~ECH DERIVED FUNCTORS AND GENERALISED
HOPF TYPE FORMULAS
Guram Donadze
A.Razmadze Mathematical Institute, Georgian Academy of Sciences
M.Alexidze St. 1, Tbilisi 380093. Georgia
donad@rmi.acnet.ge
Nick Inassaridze
A.Razmadze Mathematical Institute, Georgian Academy of Sciences
M.Alexidze St. 1, Tbilisi 380093. Georgia
inas@rmi.acnet.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 form*
*ula for
the higher homology of a group. Although substantially correct, their res*
*ult lacks
one necessary condition. We give here a counterexample to the result with*
*out that
condition. The main aim of this paper is, however, to generalise this cor*
*rected
result to derive formulae of Hopf type for the nfold ~Cech derived funct*
*ors of the
lower central series functors Zk. The paper ends with an application to a*
*lgebraic
Ktheory.
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
R \ [F, F ]
H2(G) ~=__________ ,
[F, R]
where R æ F i G is a free presentation of the group G.
Several alternative generalisations of this classical Hopf formula to higher *
*dimen
sions 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 topo
logical methods, and in particular the Hurewicz theorem for ncubes of spaces, *
*[5],
which itself is an application of the generalised van Kampen theorem for diagra*
*ms
of spaces [4]. The end result was:
___________
2000 Mathematics Subject Classification. 18G50, 18G10.
Key words and phrases. ~Cech derived functors, Hopf type formulas, Kfunctors.
1
2 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
Theorem BE ([3]). Let R1, . . . , Rn be normal subgroups of a group F such that
Q
H2(F ) = 0, Hr(F OE Ri) = 0, for r = A + 1, r = A + 2,
i2A
with AQa nonempty proper subset of = {1,Q. .,.n} (for example, if the grou*
*ps
F OE Ri are free for A 6= ) and F OE Ri~= G. Then there is an isomorph*
*ism
i2A 1 i n
n
\ Ri\ [F, F ]
Hn+1(G) ~=__i=1____________Q.
[ \ Ri, \ Ri]
A i2A i=2A
Later, Ellis using mainly algebraic means, and, in particular, his hyperrelati*
*ve
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 hopefull*
*y would
generalise further. Examining the classical case (n = 1), and the proof of the *
*usual
Hopf formula, showed a link with the ~Cech derived functors of the abelianisati*
*on
functor, (cf. [12]).
Trying to derive this result purely algebraically and to obtain Hopf type for*
*mulas
for some more general situations, we suspected that the conditions given above
for Theorem BE were not sufficient for getting the generalised Hopf formula for
Hn+1(G), n 3. In fact, we give the following counterexample to Theorem BE:
Let F be a free group with base {x1, x2}, R1, R2 and R3 normal subgroups of
the group F generated by the one point sets {x1}, {x2} and {x1x12} respectively
and G = 1. Then we have F=Ri ~=Z, i = 1, 2, 3, F=RiRj = 1, i 6= j and [F, F ] =
[R1, R2] = Ri\ Rj, i 6= j, therefore
\ Ri\ [F, F ]
__i2<3>__________
Q ~= Z
[ \ Ri, \ Ri]
A <3> i2A i=2A
whilst H3(G) = 1.
We thus set out to prove a corrected version of this BrownEllis generalised *
*Hopf
formulae, but also to generalise it further in the following direction. Homolo*
*gy
groups are the derived functors of the abelianization functor. Our generalisat*
*ion
handles the derived functors of the functors that kill higher commutators. More
precisely, let the endofunctors Zk(G) be given by Zk(G) = G= k(G), k 2, where
{ k(G), k 1} is the lower central series of a given group G. These Zk are
endofunctors on the category of groups and generalise the abelianization functo*
*r, so
their nonabelian left derived functors, LnZk, n 0, generalise the group homo*
*logy
functors Hn, n 1, cf., for instance, [1].
In [13], a Hopflike formula is proved for the second Conduch'eEllis homolog*
*y of
precrossed modules using ~Cech derived functors. The main goal of this paper is*
* to
develop this method further, and by applying it, to express LnZk, n 1, k 2,*
* by
GENERALISED HOPF TYPE FORMULAS 3
generalised Hopf type formulas. In particular, we will give the stronger condit*
*ions
needed for Theorem BE. Finally we apply these results to algebraic Ktheory.
In the first section, we introduce the ~Cech derived functors illustrating th*
*eir use
by proving the classical Hopf formula in a new way. This is not just an illustr*
*ation
as it does indicate some of the ways the argument will go later on.
In Section 2 we introduce the notion of simple normal (n + 1)ad of groups
(F ; R1, . .,.Rn) relative to Rj for some 1 j n. Then we show that for a gi*
*ven
pseudosimplicial group F*, the normal (j+1)ads of groups (Fn; Kerdn0, . .,.Ker*
*dnj1)
are simple relative to Ker dnj1for all 1 j n (Proposition 7). The main res*
*ult
of Section 2 is Theorem 9, giving that for an aspherical augmented pseudosimpli*
*cial
group (F*, d00, G), there is a natural isomorphism
n1 n1
\ Ker di \ k(Fn1)
ßnZk(F*) ~=_____i=0_______________________n1n1
Dk(Fn1; Kerd0 , . .,.Kerdn1)
for n 1, k 2. Here the Dkterm takes the form of an iterated commutator
subgroup which is an obvious generalisation of the denominator terms of both the
classical Hopf formula and the BrownEllis extension of that formula. It is al*
*so
related to the descriptions of the image term of the Moore complex, as used in
[19, 20]. The explicit formula is given at the start of Section 2.
For an inclusion crossed ncube of groups, M, given by a normal (n + 1)ad of
groups we construct, in Section 3, a new induced crossed ncube Bk(M), k 2
(Proposition 11). We show the existence of an isomorphism of simplicial groups
ZkE(n)(M)* ~= E(n)(Bk(M))*, where E(n)(M)* denotes the diagonal of the n
simplicial nerve of the crossed ncube of groups M, (Proposition 12).
Section 4 is devoted to the investigation of some properties of the mapping c*
*one
complex of a morphism of (nonabelian) group complexes introduced in [17]. In p*
*ar
ticular, for a given morphism of pseudosimplicial groups ff : G* ! H* the natur*
*al
morphism ~ : NM*(ff) ! C*(eff) induces isomorphisms of their homology groups,
where C*(eff) is the mapping cone complex of the induced morphism of the Moore
complexes and NM*(ff) is the Moore complex of a new pseudosimplicial group con
structed using ff (Proposition 13). (Here similar results have recently been f*
*ound
by Conduch'e, [8].) Using this result we derive purely algebraicly the result o*
*f [17],
(3.4. Proposition), giving for a crossed ncube of groups M an isomorphism be
tween the homotopy groups of E(n)(M)* and the corresponding homology groups
of a chain complex of groups C*(M), (Proposition 14). In particular, we give an
explicit computation of the nth homotopy group of the simplicial group E(n)(M)*
**.
In Section 5, we introduce a notion of nfold ~Cech derived functors of an en*
*do
functor on the category of groups (Theorem 16, Definition) which will be the su*
*bject
of future papers and applications to nonabelian homological algebra and Ktheor*
*y.
We give an explicit calculation of nth nfold ~Cech derived functor of the fun*
*ctor
Zk, k 2 (Theorem 20). Our method gives the possibility of finding the sought
for sufficient conditions for, and a purely algebraic proof of, the generalised*
* Hopf
4 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
formula of Brown and Ellis, moreover we express LnZk(G), n 1, k 2 by a Hopf
type formula (Theorem 21).
In Section 6, an application to algebraic Ktheory is given. In particular, Q*
*uillen's
algebraic Kfunctors Kn+1, n 1 are described in terms of short exact sequence
including the higher Hopf type formulae for free exact npresentations induced *
*by a
free simplicial resolution of the general linear group (Theorem 25).
1. An approach to the classical Hopf formula via ~Cech derived
functors
We give here a brief introduction to ~Cech derived functors. A fuller accoun*
*t is
given in [12]. We will limit ourselves to the ~Cech derived functors of the Abe*
*lian
ization functor. Later we will develop the nfold analogue of some of this theo*
*ry.
Definition. Let T : Gr ! Gr be a covariant functor. Define ith C~ech deri*
*ved
functor LiT : Gr ! Gr , i 0, of the functor T by choosing for each G in Gr ,*
* a
ff
free presentation F : R æ F i G of G and setting
LiT (G) = ßi(T ~C(ff)*) ,
where (C~(ff)*, ff, G) is the ~Cech resolution of the group G for the free pres*
*entation
F of G. This latter resolution is constructed as follows:
Given a group G and a homomorphism of groups ff : F ! G. The ~Cech augmented
complex (C~(ff)*, ff, G) for ff is
dn0 d20 1
!. !. ! d0! ff
. . ...F xG . .x.GF ... .!. F xG F ! F ! G ,
! !dn ! d1
n d22 1
thus
~C(ff)n = F xG . .x.GF = {(x , . .,.x ) 2 F n+1 ff(x ) = . .=.ff(x )} for n *
* 0 ,
______z_____" 0 n 0 n
(n+1)times
dni(x0, . .,.xn) = (x0, . .,.^xi, . .,.xn)
and
sni(x0, . .,.xn) = (x0, . .,.xi, xi, xi+1, . .,.xn)
for 0 i n (see [12]).
In case F is a free group and ff is an epimorphism as in F above, (C~(ff)*, f*
*f, G)
will be called a ~Cech resolution of G. Example. The prime example of the ~Ce*
*ch
derived functors are those of the abelianization functor. We recall that
H2(G) ~=L1Ab (G)
and will use this later.
Now using crossed modules and their nerves, we present a fresh view of the ~C*
*ech
complex which leads to some ideas that will be useful throughout the paper.
GENERALISED HOPF TYPE FORMULAS 5
First, let us recall the definition of crossed module. A crossed module (M, P*
*, ~) is
a group homomorphism ~ : M ! P together with a (left) action of P on M which
satisfies the following conditions:
(i)~(pm) = p~(m)p1,
(ii)~(m)m0= mm0m1, (Peiffer identity)
for all m, m02 M and p 2 P .
A morphism (', _) : (M, P, ~) ! (N, Q, ) of crossed modules is a commutative
square of groups
'
M _____//N
~  
fflfflfflffl_
P _____//Q
such that '(pm) = _(p)'(m) for all m 2 M, p 2 P . Let us denote the category of
crossed modules by CM .
It is well known from [17] that the category CM of crossed modules is equiva*
*lent
to the category of cat1groups (for the definition see [17]), and for a given c*
*rossed
~ 1
module, M = (M ! P ), the corresponding cat group is (M o P, s, t), where
s(m, p) = p and t(m, p) = ~(m)p. This cat1group has an internal category stuct*
*ure
within the category Gr of groups and the nerve of its category structure forms *
*the
following simplicial group
____//_. _____//. _d0_//_ d0
E(M)* : . ._...//_E(M)n____..//_. . .___//_//_E(M)1__////_E(M)0,
d2 d1
where E(M)n = M o (. .(.M o P ) . .).with n semidirect factors of M, and face
and degeneracy homomorphisms are defined by
d0(m1, . .,.mn, p) = (m2, . .,.mn, p) ,
di(m1, . .,.mn, p) = (m1, . .,.mimi+1, . .,.mn, p) , 0 < i < n ,
dn(m1, . .,.mn, p) = (m1, . .,.mn1, ~(mn)p) ,
si(m1, . .,.mn, p) = (m1, . .,.mi, 1, mi+1, . .,.mn, p) , 0 i n .
The simplicial group E(M)* is called the nerve of the crossed module M and its
Moore complex is trivial in dimensions 2. In fact its Moore complex is just
the original crossed module up to isomorphism with M in dimension 1 and P in
dimension 0.
Lemma 1. Let G be a group and F !ff G be a homomorphism of groups. Then
the ~Cech complex for ff is isomorphic to the nerve of the inclusion crossed mo*
*dule
E(R ,! F )*, where R = Ker ff.
6 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
Proof.Let us compare the following two simplicial groups, the ~Cech complex for*
* ff
and the nerve of the inclusion crossed module R ,! F
______//. ______//_
E(R ,! F )* : . ._...__//R o R o F______//_//_R o_F__//_//_F

~2 ~1 ~0
____//_ fflffl_____// fflffl fflffl
C~(ff)* : . . ....F x F x F _____//F x F_____////_F
____//_ G G _____// G
by constructing a morphism ~* as follows:
~0 is the identity on F ; and
~n(r1, . .,.rn, f) = (r1. .r.nf, r2. .r.nf, . .,.rnf, f) for all n 1 *
*and
(r1, . .,.rn, f) 2 E(R ,! F )n.
It is easy to check that ~* is isomorphism of simplicial groups.
We recall from [16] that a crossed module ~ : 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 al*
*so
abelian. Let us denote the category of abelian crossed modules by AbCM .
One can define the abelianization functor Ab from the category CM to the cat*
*e
~
gory AbCM in the following way: for any crossed module M = (M ! P ),
M _~ P
Ab(M) = (______ ! ______) ,
[P, M] [P, P ]
in which [P, M] is the subgroup of M generated by the elements pmm1 for all
m 2 M, p 2 P and __~is induced by ~.
Given a simplicial group G*, let us apply the group abelianization functor di*
*mension
wise, denote the resulting simplicial group by Ab (G*).
~
Proposition 2. Let M ! P be a crossed module. Then there is an isomorphism of
simplicial groups
~ ~
Ab(E(M ! P )*) ~=E(Ab (M ! P ))* .
Proof.Let us consider the two simplicial groups
~ ____//_. ab___//_ ___ab//_
Ab(E(M ! P )*) : : . ._...//_(M o M o P ) ____//_//_(M o P_)_//_P ab
and
~ ____//_.M M ab____//_M ab____//_
E(Ab (M ! P ))* : : . ._._..//___ x ______ x P ____//_//__ x P ____//*
*_P ab.
[P, M] [P, M] [P, M]
It is easy to show that
M ab
(M o P )ab ~! ______x P ,
[P, M]
~[(m, p)] = ([m], [p])
and
_M____ ab~0 ab
x P ! (M o P ) ,
[P, M]
GENERALISED HOPF TYPE FORMULAS 7
~0([m], [p]) = [(m, p)]
are homomorphisms and ~~0= ~0~ = 1. Using these isomorphisms one has
M ab
(M o (M o P ))ab ~= ___________x (M o P )
[M o P, M]
~= ____M______x __M___ x P ab
[M o P, M] [P, M]
~= _M____x __M___ x P ab,
[P, M] [P, M]
since [M o P, M] = [P, M] as
(m,p)m0m01 = mpm0m1m01
= mpm0m01m1mm0m1m01
= m(pm0m01)m1(~(m)m0m01) 2 [P, M]
for all m, m02 M, p 2 P . It is also easy to see by the same sort of argument t*
*hat
there exist isomorphisms between higher terms of these simplicial groups and th*
*at
these isomorphisms are compatible with face and degeneracy maps.
Now we are ready to revisit the classical Hopf formula.
One can prove the Hopf formula in many ways, but for our later generalization
we will prove it using the ~Cech derived functors.
i
Theorem 3. Let G be a group and R æ F i G be a free presentation of the group
G. Then there is an isomorphism of groups
R \ [F, F ]
H2(G) ~=__________ .
[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 ~Cech derived functor of the group abelianization funct*
*or.
Now Lemma 1 and Proposition 2 implies an isomorphism
i
H2(G) ~=ß1(E(Ab (R ,! F ))*) .
As we already mentioned above, it is clear that ß1(E(Ab (R ,! F ))*) is isomo*
*rphic
to the kernel of the crossed module
` _ '
R i F
Ab(R ,! F ) = ______! ______
[F, R] [F, F ]
giving the desired result.
8 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
ff
Corollary 4. Let R æ K i G be a presentation of a group G and H2(K) = 0.
Then there is an isomorphism of groups
R \ [K, K]
H2(G) ~=___________.
[K, R]
fi
Proof.Consider a free presentation R00æ F i K of the group K. Hence, one has
fffi
also a free presentation R0æ F i G of the group G. It is easy to check that th*
*ere
is an exact sequence of groups
R00\_[F,_F_] R0\ [F, F ] R K G
! __________ ! ______ ! ______ ! ______ ! 1 ,
[F, R00] [F, R0] [K, R] [K, K] [G, G]
Thus by Theorem 3 and the condition that H2(K) = 0, one has the following exact
sequence of groups
R ab ab
0 ! H2(G) ! ______ ! K ! G ! 1 ,
[K, R]
which completes the proof.
2.Simple normal (n + 1)ad of groups
One of the tools we will be using later is the theory of crossed ncubes of g*
*roups.
These generalise normal (n + 1)ads of groups in the same way that crossed modu*
*les
generalise normal subgroups. We therefore start by developing some techniques f*
*or
handling (n + 1)ads of groups, relating them to iterated commutators.
Given a group F and n normal subgroups, R1, . .,.Rn, then (F ; R1, . .,.Rn) w*
*ill be
called a normal (n + 1)ad of groups. A normal (n + 1)ad of groups (F ; R1, . *
*.,.Rn)
is called simple relative to Rj for some 1 j n if there exists a subgroup F*
* 0of
the group F such that
F 0\ Rj = 1 , \ Ri= ( \ Ri\ F 0)( \ Ri\ Rj)
i2A i2A i2A
for all A \ {j}.
For a given (n + 1)ad of groups (F ; R1, . .,.Rn), A and k 1 denote *
*by
Dk(F ; A) the following normal subgroup of the group F
Y
[ \ Ri, [ \ Ri, . . ., [ \ Ri, \ Ri] . .].] .
A1[A2[...[Ak=Ai2A1 i2A2 i2Ak1 i2Ak
Sometimes we write Dk(F ; R1, . .,.Rn) instead of the notation Dk(F ; ) .
Lemma 5. Let (F ; R1, . .,.Rn) be a normal (n + 1)ad of groups which is simple
relative to Rj, 1 j n and k 1, then
Dk(F ; A) = (Dk(F ; A) \ F 0)Dk(F ; A [ {j})
for all A \ {j}.
GENERALISED HOPF TYPE FORMULAS 9
Proof.We use induction on k. Let k = 1, then
D1(F ; A) = \ Ri= ( \ Ri\ F 0)( \ Ri\ Rj) = ( \ Ri\ F 0)D1(F ; A [ {j})
i2A i2A i2A i2A
for A \ {j}.
Proceeding by induction, we suppose that the assertion is true for k  1 and *
*we
will prove it for k.
The inclusion (Dk(F ; A) \ F 0)Dk(F ; A [ {j}) Dk(F ; A) is obvious. It is*
* easy
to see that a generator of Dk(F ; A) has the form [x, w], where x 2 \ Ri, w 2
i2B
Dk1(F ; C), B, C A \ {j} and B [ C = A. There exist elements y 2
\ Ri\ F 0and z 2 \ Ri\ Rj such that x = yz. One has
i2B i2B
[x, w] = [yz, w] = y[z, w]y1[y, w] .
Clearly [z, w] 2 Dk(F ; B [ C [ {j}) = Dk(F ; A [ {j}) and hence y[z, w]y1 2
Dk(F ; A [ {j}). By inductive hypothesis there exist w0 2 Dk1(F ; C [ {j}) and
x02 Dk1(F ; C) \ F 0such that w = x0w0. One has
[y, w] = [y, x0w0] = [y, x0]x0[y, w0]x01.
Clearly [y, w0] 2 Dk(F ; B [ C [ {j}) = Dk(F ; A [ {j}) and hence x0[y, w0]x01*
* 2
Dk(F ; A [ {j}). Therefore there is an element w002 Dk(F ; A [ {j}) such that
[x, w] = [y, x0]w00where [y, x0] 2 Dk(F ; A) \ F 0.
For a given group G the (lower) central series k = k(G),
G = 1 2 . . . k . . .
Q
of G is defined inductively by k = [ i, j]. The wellknown WittHall iden*
*tities
i+j=k
on commutators (see e.g. [2]) imply that k = [G, k1].
Let Gr denote the category of groups. Let us define higher abelianization t*
*ype
functors Zk : Gr ! Gr , k 2 by Zk(G) = G= k(G) for any G 2 ob Gr and
where Zk(ff) is the natural homomorphism induced by a group homomorphism ff.
Of course, Z2 is the ordinary abelianization functor of groups.
Proposition 6. Let (F ; R1, . .,.Rn) be a normal (n + 1)ad of groups and k 2.
Suppose that (F ; R1, . .,.Rj) is a simple normal (j + 1)ad of groups relative*
* to Rj
for all 1 j n. Then
\ Ri\ k(F ) = Dk(F ; ) , 1 j n .
i2
Proof.Since the inclusion Dk(F ; ) \ Ri\ k(F ) is clear, we only have to *
*show
i2
the inclusion \ Ri\ k(F ) Dk(F ; ), which will be done by induction on j.
i2
Let j = 1, then there exists a subgroup F1 of the group F such that R1 \ F1 =*
* 1
and F = F1R1. Let w 2 R1 \ k(F ) k(F ) = Dk(F ; ;). Using Lemma 5 one
has elements x0 2 Dk(F ; ;) \ F1 and w0 2 Dk(F ; <1>) such that w = x0w0. But
01 0
x0= ww 2 R1 and hence x = 1. Thus R1 \ k(F ) Dk(F ; <1>).
10 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
Proceeding by induction, we suppose that the result is true for j  1 and we *
*will
prove it for j.
There exists a subgroup Fj of the group F such that Rj \ Fj = 1 and \ Ri =
i2A
( \ Ri \ Fj)( \ Ri \ Rj) for all A {1, . .,.j  1}. Let w 2 \ Ri \ k(F )
i2A i2A i2
\ Ri\ k(F ). Using the inductive hypothesis one has the equality \ Ri\
i2 i2
k(F ) = Dk(F ; ). By Lemma 5 there are elements x02 Dk(F ; ) \ Fj
01
and w0 2 Dk(F ; ) such that w = x0w0. Certainly x0 = ww 2 Rj and hence
x0= 1. Therefore \ Ri\ k(F ) Dk(F ; ).
i2
These conditions may seem rather restrictive, but the following observation s*
*hows
that examples of simple normal (n + 1)ads of groups appear naturally, and that
moreover these examples satisfy the conditions of Proposition 6. First some ter*
*mi
nology and notation on pseudosimplicial groups, (cf. [12] for the general theor*
*y).
A pseudosimplicial group G* is a nonnegatively graded group with face homo
morphisms dni: Gn ! Gn1 and pseudodegeneracies, sni: Gn ! Gn+1, 0 i n,
satisfying all the simplicial identities except possibly the identity sn+1isnj=*
* sn+1j+1sni
for i j (again, see [12]). The Moore complex (NG*, @*) of G* is the chain com
n1
plex defined by NGn = \ Ker dniwith @n : NGn ! NGn1 induced from dnnby
i=0
restriction. The homotopy groups of G* are defined as the homology groups of the
complex (NG*, @*), i.e. ßn(G*) = Hn(NG*, @*), n 0 (see [12]). For good exampl*
*es
of pseudosimplicial groups see [18].
Proposition 7. Let F* be a pseudosimplicial group. Then (Fn; Kerdn0, . .,.Kerdn*
*j1)
is a simple normal (j + 1)ad of groups relative to Ker dnj1for all 1 j n.
Proof.Since dnj1sn1j1= 1, sn1j1(Fn1) \ Ker dnj1= 1 and sn1j1(Fn1) Ker*
*dnj1= Fn
for all n 1. Hence when j = 1, (Fn; Kerdn0) is a simple normal 2ad of groups
relative to Ker dn0and the F 0of the definition of simplicity is sn10(Fn1).
Now suppose that j > 1. We will show the following equality
\ Ker dni= ( \ Ker dni\ sn1j1(Fn1))( \ Ker dni\ Ker dnj1)
i2A i2A i2A
for all A {0, . .,.j  2} and A 6= ;, so again the F 0of the definition of si*
*mplicity
is sn1j1(Fn1). Let x = sn1j1(xn1)rj1 2 \ Ker dni, where xn1 2 Fn1, r*
*j1 2
i2A
Ker dnj1. Thus dni(x) = dnisn1j1(xn1)dni(rj1) = 1 for all i 2 A. Since i*
* < j  1,
one has dni(rj1) = sn2j2dn1i(xn1)1. Hence 1 = dn1idnj1(rj1) = dn1j2d*
*ni(rj1) =
dn1j2sn2j2dn1i(xn1)1 = dn1i(xn1)1. Therefore dni(rj1) = 1 and dnisn*
*1j1(xn1) =
1 for all i 2 A.
The next lemma is well known, but very useful. The proof is routine.
Lemma 8. Let G* be a pseudosimplicial group and A , A 6= , then
dnn( \ Ker dni1) = \ Ker dn1i1, n 2.
i2A i2A
GENERALISED HOPF TYPE FORMULAS 11
Next let us consider an augmented pseudosimplicial group (F*, d00, G) and, ap*
*ply
ing the functor Zk dimensionwise, denote the resulting augmented pseudosimplic*
*ial
group by (Zk(F*), Zk(d00), Zk(G)). Our previous results will allow calculation *
*of the
homotopy groups of Zk(F*) in certain important cases notably the following.
Theorem 9. If (F*, d00, G) is aspherical then there is a natural isomorphism
n1 n1
\ Ker di \ k(Fn1)
ßnZk(F*) ~=_____i=0________________________n1n1, k 2, n 1 .
Dk(Fn1; Kerd0 , . .,.Kerdn1)
Proof.Let us consider the short exact sequence of augmented pseudosimplicial
groups
1 1 1 1
# # # #
fdn0 f1
!. !. ! !d0 fd00
. . ... k(Fn) ... .!. k(F1) ! k(F0) ! k(G)
! ! ! f1
fdnn d1
# # # #
dn0 1
!. !. ! !d0 d00 .
. . ... Fn ... .!. F1 ! F0 ! G
! !dn ! d1
n 1
# # # #
!. !. !
. . ...Zk(Fn) ... .!. Zk(F1) !! Zk(F0) ! Zk(G)
! ! !
# # # #
1 1 1 1
By the induced long exact homotopy sequence, one has the isomorphisms of groups
n1 gn1
\ Kerdi
ßnZk(F*) ~=_i=0_________n1, n 1 .
denn( \ Kerden)
i=0 i
Since deniis a restriction of dnito k(Fn), Ker deni= Kerdni\ k(Fn). Hence
n1 gn1 n1 n1 n1 n1
\ Ker di = ( \ Kerdi ) \ k(Fn1) and \ Ker edni= ( \ Kerdni) \ k(Fn).
i=0 i=0 i=0 i=0
Using Proposition 6 and Proposition 7 one has
( \ Ker dni1) \ k(Fn) = Dk(Fn; Kerdn0, . .,.Kerdnn1)
i2
for n 1.
Since (F*, d00, G) is an aspherical augmented pseudosimplicial *
*group,
dnn( \ Ker dni1) = \ Ker dn1i1, n 1. Using this fact and Lemma 8, it i*
*s now
i2 i2
easy to see that one has an equality
ednn(n1\Kerden) = dn(Dk(Fn; Kerdn, . .,.Kerdn )) = Dk(Fn1; Kerdn1, . .,.Ker*
*dn1) .
i=0 i n 0 n1 0 *
* n1
12 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
In the case where (F*, d00, G) is a free pseudosimplicial resolution of G, th*
*ese homo
topy groups will be the left nonabelian derived functors LnZk(G) of Zk, evalua*
*ted
at G. (For more on these nonabelian derived functors, we refer the reader to [*
*12].)
We thus have the following formal result.
Corollary 10. Let G be a group and (F*, d00, G) an aspherical augmented pseudos*
*im
plicial group and k 2. If Fn is a free group for all n 0, i.e. (F*, d00, G)*
* is a free
pseudosimplicial resolution of the group G, then there is a natural isomorphism
n1 n1
\ Ker di \ k(Fn1)
LnZk(G) ~=_____i=0_______________________n1n1, n 1 .
Dk(Fn1; Kerd0 , . .,.Kerdn1)
Proof.Straightforward from Theorem 9.
3. Crossed ncubes of groups
The following definition is due to Ellis and Steiner [11] (see also [25]). A *
*crossed n
cube of groups is a family {MA : A } of groups, together with homomorphis*
*ms,
~i : MA ! MA\{i}for i 2 , A and functions h : MA x MB  ! MA[B
for A, B , such that if ab denotes h(a, b) . b for a 2 MA and b 2 MB with
A B, then for all a, a02 MA, b, b02 MB , c 2 MC and i, j 2 , the following
conditions hold:
~i(a) = a if i =2A,
~i~j(a) = ~j~i(a),
~ih(a, b) = h(~i(a), ~i(b)),
h(a, b) = h(~i(a), b) = h(a, ~i(b)) if i 2 A \ B,
h(a, a0) = [a, a0],
h(a, b) = h(b, a)1,
h(a, b) = 1 if a = 1 or b = 1,
h(aa0, b) = ah(a0, b)h(a, b),
h(a, bb0) = h(a, b)bh(a, b0),
ah(h(a1, b), c)ch(h(c1, a), b)bh(h(b1, c), a) = 1,
ah(b, c) = h(ab,a c) if A B \ C.
A morphism of crossed ncubes, {MA} ! {M0A} , is a family of group homomor
phisms {fA : MA ! M0A, A } commuting with the ~i and the hfunctions.
This gives a category of crossed ncubes of groups which will be denoted by Crs*
*n.
GENERALISED HOPF TYPE FORMULAS 13
Examples.
(i)A crossed 1cube is the same as a crossed module (see Section 1).
(ii)A crossed 2cube is the same as a crossed square (for the definition see*
* [4]).
The detailed reformulation is easy.
(iii)Let G be a group and N1, . .,.Nn be normal subgroups of G. Let MA =
\ Ni for A (we also note that then M; = G); if i 2 , define
i2A
~i: MA i2A!MA\{i}to be the inclusion and given A, B , let h : MA x
MB ! MA[B be given by the commutator: h(a, b) = [a, b] for a 2 MA, b 2
MB (here, of course, MA[B = MA \ MB ) . Then {MA : A , ~i, h}
is a crossed ncube, called the inclusion crossed ncube given by the no*
*rmal
(n + 1)ad of groups (G; N1, . .,.Nn).
Ellis and Steiner [11] prove that Crsn is equivalent to the category of catn*
*groups
introduced by Loday who proved that equivalence for n = 1, 2, [17].
For a given crossed ncube M, there is an associated catngroup and hence on
applying the crossed module nerve structure E, examined in Section 1, in the n
independent directions, this construction leads naturally to an nsimplicial gr*
*oup,
called the multinerve of the crossed ncube M and denoted by Ner(M). Taking the
diagonal of this nsimplicial group gives a simplicial group denoted by E(n)(M)*
**,
(see [25]).
Proposition 11. Let M be an inclusion crossed ncube given by a normal (n+1)ad
of groups (F ; R1, . .,.Rn) and k 2. Then there is a crossed ncube Bk(M) giv*
*en
by:
(a) for A
Bk(M)A = \ RiOEDk(F ; A) .
i2A
(b) if j 2 , the homomorphism e~j: Bk(M)A ! Bk(M)A\{j}is induced by the
inclusion homomorphism ~j.
(c) representing an element in Bk(M)A by __xwhere x 2 \ Ri (the bar denotes*
* a
i2A
coset), and for A, B ,_the_map_eh:_Bk(M)A_x Bk(M)B ! Bk(M)A[B
is given by eh(__x, __y) = h(x, y)= [x, y]for all __x2 Bk(M)A, __y2 Bk(M*
*)B .
Proof.In our notation
Y
Dk(F ; A) = [ \ Ri, [ \ Ri, . .,.[ \ Ri, \ Ri] . .].] , A*
* .
A1[A2[...[Ak=Ai2A1 i2A2 i2Ak1 i2Ak
Since
[ \ Ri, [ \ Ri, .,.[. \ Ri, \ Ri] . .].]
i2A1 i2A2 i2Ak1 i2Ak
[ \ Ri, [ \ Ri, . .,.[ \ Ri, \ Ri] . *
*.].]
i2A1\{j} i2A2\{j} i2Ak1\{j} i2Ak\{j}
14 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
for A1 [ . .[.Ak = A , the inclusion ~j : \ Ri ,! \ Ri induces the
i2A i2A\{j}
homomorphism e~j: Bk(M)A ! Bk(M)A\{j}for all j 2 .
Now we are only left to show that the function eh : Bk(M)A x Bk(M)B !
Bk(M)A[B for A, B is well defined. In fact, let x0 2 \ Ri, y0 2 \ Ri *
*be
i2A i2B
such that
Y
xx012 [ \ Ri, [ \ Ri, . .,.[ \ Ri, \ Ri] . .].]
A1[...[Ak=Ai2A1 i2A2 i2Ak1 i2Ak
and Y
yy012 [ \ Ri, [ \ Ri, . .,.[ \ Ri, \ Ri] . .].].
A1[...[Ak=Bi2A1 i2A2 i2Ak1 i2Ak
The inclusion
[ \ Ri, \ Ri] \ Ri
i2A i2B i2A[B
for all A, B implies that
[x, y][x0, y0]1= xyx1y1y0x0y01x01
= xy0[y01y, x1]y01x1x[y0, x1x0]x1
Y
2 [ \ Ri, [ \ Ri, . .,.[ \ Ri, \ Ri] . .]*
*.] ,
A1[...[Ak=A[Bi2A1 i2A2 i2Ak1 i2Ak
_______ ________
so h(x, y)= h(x0, y0)and ehis well defined. The verification that Bk(M) is a cr*
*ossed
ncube is routine and is left as an exercise.
Remark. The functor B2 coincides on the subcategory of inclusion crossed ncubes
with the abelianization functor Ab from the category Crsn to the category AbCrs*
* n
of abelian crossed ncubes of groups (i.e. crossed ncubes all of whose h maps *
*are
trivial), considered for n = 1 in Section 1 and defined in general by the follo*
*wing
way: for any M = {MA : A , ~i, h} in Crs n, Ab (M) is an abelian crossed
ncube given by:
(a) for A
MA
Ab(M)A = ____________Q,
DB,C
B,C
B[C=A
where DB,C is the subgroup of MA generated by the elements h(b, c),
h : MB x MC ! MB[C=A for all b 2 MB , c 2 MC .
(b) if i 2 , the homomorphism e~i: Ab(M)A ! Ab(M)A\{i}is induced by the
homomorphism ~i.
(c) for A, B , the function eh : Ab (M)A x Ab (M)B ! Ab (M)A[B is
induced by h and therefore is trivial.
The functor Ab : Crsn ! AbCrs n is left adjoint to the inclusion fu*
*nctor
i : AbCrs n,! Crsn as is easily checked.
GENERALISED HOPF TYPE FORMULAS 15
For any inclusion crossed ncube M given by a normal (n + 1)ad of groups
(F ; R1, . .,.Rn) and k 2, there is a natural morphism of crossed ncubes M !
n,k*(n)
Bk(M) inducing the natural fibration of simplicial groups E(n)(M)* ! E (Bk(*
*M))*
defined by
n,km(x1, . .,.xl) = (___x1, . .,.__xl)
for all (x1, . .,.xl) 2 E(n)(M)m = ( \ Ri) o . .o.( \ Ri) and m 0, where
i2A1 i2Al
A1, . .,.Al and l = (m + 1)n. It is easy to see that Ker n,km= Dk(F ; A1*
*) o
. .o.Dk(F ; Al).
Let us consider a simplicial group G* and, applying the functor Zk dimension*
*wise,
denote the resulting simplicial group by ZkG*.
Proposition 12. Let M be an inclusion crossed ncube given by a normal (n+1)ad
of groups (F ; R1, . .,.Rn) and k 2. Then there is an isomorphism of simplic*
*ial
groups
ZkE(n)(M)* ~=E(n)(Bk(M))* .
Proof.For any inclusion crossed module R ,! F , It is easy to check the followi*
*ng
equalities in the group R o . .o.R o F :
[(1, . .,.1, x), (1, . .,.1, x0)] = (1, . .,.1, [x, x0]) ,
[(1, . .,.1, r, 1, . .,.1), (1, . .,.1, x)] = (1, . .,.1, [r, x], 1, . .,*
*.1) ,
s s
[(1, . .,.1, r, 1, . .,.1), (1, . .,.1, r0, 1, . .,.1)] = (1, . .,.1, [r,*
* r0], 1, . .,.1)
s t min*
*{s, t}
for all x, x02 F , r, r02 R.
There are further generalisation of these equalities, namely for any inclusion
crossed ncube M given by the normal n + 1ad of groups (F, R1, . .,.Rn) one has
the following facts, the proof of which is routine and will be omitted.
(A) Let s and t be any fixed elements of the set <(m + 1)n>. Then there exis*
*ts
a unique ~ = ~(s, t) 2 <(m + 1)n> such that A~ = As [ At and in the group
E(n)(M)m there holds the equality:
[(1, . .,.1, x, 1, . .,.1), (1, . .,.1, y, 1, . .,.1)] = (1, . .,.1, [x,*
* y], 1, . .,.1)
s t ~
for all x 2 \ Ri, y 2 \ Ri.
i2As i2At
(B) Let s 2 <(m + 1)n> and A, B As with A [ B = As. Then there exists
p, q 2 <(m + 1)n> such that Ap = A, Aq = B and ~(p, q) = s.
We only have to show the equality
(1) k(E(n)(M)m ) = Ker n,km
which will be done by induction on k, using facts (A) and (B) above.
Let k = 1, then it is clear that 1(E(n)(M)m ) = Ker n,1m.
Proceeding by induction, we suppose that (1) is true for k  1 and we will pr*
*ove
it for k.
16 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
First we will show the inclusion Ker n,km k(E(n)(M)m ). It is sufficient t*
*o show
1 o . .o.1 o Dk(F, As) o 1 o . .o.1 k(E(n)(M)m ) for all s 2 <(m + 1)n> .
In fact, any generator w of Dk(F, As) has the form w = [x, y], where x 2 \ Ri,
i2A
y 2 Dk1(F, B) and A [ B = As.
Now (B) implies that there exist p, q 2 <(m + 1)n> such that Ap = A, Aq = B a*
*nd
~(p, q) = s. Thus we have
[(1, . .,.1, x, 1, . .,.1), (1, . .,.1, y, 1, . .,.1)] = (1, . .,.1, w, *
*1, . .,.1) ,
p q s
which means that
1 o . .o.1 o Dk(F, As) o 1 o . .o.1 [E(n)(M)m , Ker n,k1m] .
Therefore by the inductive hypothesis we obtain
1o. .o.1oDk(F, As)o1o. .o.1 [E(n)(M)m , k1(E(n)(M)m )] = k(E(n)(M)m ) .
Finally we will show the inverse inclusion k(E(n)(M)m ) Ker n,km. In fac*
*t,
any generator w of k(E(n)(M)m ) could be written as w = [w1, w2], where w1 2
E(n)(M)m and w2 2 k1(E(n)(M)m ). Using again the inductive hypothesis we
have w2 2 Ker n,k1m. Thus
(m+1)nY
w1 = (1, . .,.1, xs, 1, . .,.1) , xs 2 \ Ri,
s=1 s i2As
(m+1)nY
w2 = (1, . .,.1, yt, 1, . .,.1) , yt 2 Dk1(F, At) .
t=1 t
It is certain that [xs, yt] 2 Dk(F, As [ At). Then (A) implies that we have
[(1, . .,.1, xs, 1, . .,.1), (1, . .,.1, yt, 1, . .,.1)] = (1, . .,.1, [xs*
*, yt], 1, . .,.1)
s t ~(*
*s,t)
2 1 o . .o.1 o Dk(F, A~(s,t)) o 1 o . .o.1 Ker n,km.
Then the WittHall identities on commutators implies that w 2 Ker n,km.
4.Nonabelian mapping cone complex
A complex of (nonabelian) groups (A*, d*) of length n is a sequence of group
homomorphisms
dn1 d1
An dn!An1 ! . ..! A0
such that Im di+1 is normal in Ker di. Now we recall the following definition *
*from
[17].
Let f : (A*, d*) ! (B*, d0*) be a morphism of chain complexes of groups. Let*
* f
satisfy the following conditions (*):
each fi: Ai! Bi is a crossed module
and
GENERALISED HOPF TYPE FORMULAS 17
the maps (di, d0i) form a morphism of crossed modules.
Then the mapping cone of f is a complex of (nonabelian) groups (C*(f), @*)
defined by Ci(f) = Ai1o Bi, where the action of Bi on Ai1 is induced by the
action of Bi1 on Ai1 via the homomorphism d0i; and
@i(a, b) = (di1(a)1, fi1(a)d0i(b))
for all a 2 Ai1, b 2 Bi. Then by [17], Proposition 3.2, there is a long exact *
*sequence
of groups.
(2) . ..! Hi(A*) ! Hi(B*) ! Hi(C*(f)) ! Hi1(A*) ! . ...
Now let us consider a morphism of pseudosimplicial groups ff : (G*, d*i, s*i)*
* !
(H*, d0*i, s0*i) satisfying conditions (* *) :
each ffn : Gn ! Hn is a crossed module
and
the maps (d*i, d0*i) and (s*i, s0*i) form morphisms of crossed modules.
Define a new pseudosimplicial group M*(ff) in the following way:
Mn(ff) = Gn_o_Gn_o_._.o.Gnz________"oHn ,
ntimes
dn0(g1, . .,.gn, h) = (dn0(g2), . .,.dn0(gn), d0n0(h)) ,
dni(g1, . .,.gn, h) = (dni(g1), . .,.dni(gi)dni(gi+1), . .,.dni(gn), d0ni(h)) *
*, 0 < i < n ,
dnn(g1, . .,.gn, h) = (dnn(g1), . .,.dnn(gn1), ffn1dnn(gn)d0nn(h)) ,
sni(g1, . .,.gn, h) = (sni(g1), . .,.sni(gi), 1, sni(gi+1), . .,.sni(gn), s0ni*
*(h)) , 0 i n .
It is easy to see that the induced morphism eff: NG* ! NH*, where NG* and
NH* are the Moore complexes of G* and H* respectively, satisfies the conditions
(*). Therefore one can consider the mapping cone complex C*(eff) of eff.
Proposition 13. The natural morphism of complexes ~ : NM*(ff) ! C*(eff), given
by ~n(g1, g2, . .,.gn, h) = (dnn(gn), h), n 0 induces an isomorphism of groups
ßn(M*(ff)) ~=Hn(C*(eff)) , n 0 .
Proof.The verification that ~n, n 0 is a homomorphism and commuting with
differentials is easy. Let (g, h) 2 NGn1 o NHn = Cn(eff), then it is easy to c*
*heck
that (sn10(g)ffl(n1), . .,.sn1n2(g)1, sn1n1(g), h) 2 NMn(ff), where ffl(*
*i) = (1)i. It is
clear that ~n(sn10(g)ffl(n1), . .,.sn1n2(g)1, sn1n1(g), h) = (g, h). Hen*
*ce ~n is surjective
for all n 0. __ __
Consider_the kernel complex (G*, @*) of ~. Note that Im @n is not normal in
Ker @n1 in general, G0 = 1 and
8 9
>>  dn0(gj) = 1 , 2 j n ; >
< (g1, g2, . .,.gn)2 dn(g ) = dn(g )dn(g ) = 1 , 1 i n >1=,
Gn = Gn o Gn o . .o.Gn i j i i i i+1 .
>>_________z________" 1 j n and i 6= j  1, j>;
: ntimes  dn >;
n(gn) = 1
18 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
_____
Furthermore, it is easy to check that for an element (g1, . .,.gn1) 2 Ker @n1*
*, the
element (g01, . .,.g0n1, g0n), defined by the formulae:
(
sn1n1(gi)sn1n2(g1i) . .s.n1i(gffl(ni1)i)sn1i1(gffl(ni)igffl*
*(ni)i+1.g.f.fl(ni)n1), i even,
g0i= n1 ffl(ni) ffl(ni)ffl(ni)n1ffl(ni1) n1 1 n1
si1(gn1 . .g.i+1 gi )si (gi ) . .s.n2(gi )sn1(gi), i odd,
for all 1 i n  1 and g0n= 1, belongs to Gn and
___ 0 0 0
@n(g1, . .,.gn1, gn) = (g1, . .,.gn1).
Now the proposition follows from the long exact homology sequence induced by the
short exact sequence of complexes 1 ! G* ! NM*(ff) ~! C*(eff) ! 1.
Given a pseudosimplicial group G*, we will say that the length of G* is n,
denoted by l(G*) n if NGi= 1 for i > n.
Remark. Let ff : (G*, d*i, s*i) ! (H*, d0*i, s0*i) be a morphism of pseudosimp*
*licial
groups satisfying the conditions (**) and n 2. Suppose l(G*) n  1 and
l(H*) n  1. Consider an element (g1, g2, . .,.gk, h) 2 NMk(ff), k > n, then
dk0(gj) = 1 , 2 j k ,
dki(gj) = dki(gi)dki(gi+1) = 1 , 1 i k  1 , 1 j k and i 6= j  *
*1 , j ,
d0ki(h) = 1 , 0 i k  1.
By Lemma 8 one can easily show that gi = 1, 1 i k and h = 1, meaning
NMk(ff) = 1 for k > n. Thus l(M*(ff)) n.
Now using the mapping cone construction, for a given crossed ncube M, we
construct inductively a complex of groups C*(M) of length n, always having in
mind that M is thought as a crossed module of crossed (n  1)cubes, M1 ! M0
~
(Proposition 5, [25]). In fact, for n = 1, and M = (M ! P ), C*(M) is the
complex M ! P of length 1. Let n = 2 and M be a crossed square, considered
as a crossed 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 2crossed module structure, [6], as noted by Conduch'e, *
*see
also [21].) Proceeding by induction, suppose for any crossed (n1)cube M we ha*
*ve
constructed a complex C*(M) of length n  1. Now let M be a crossed ncube and
consider it as a crossed module of crossed (n  1)cubes M1 ! M0, which implies
a morphism of complexes of groups C*(M1) ffi!C*(M0) of length n  1 satisfying
the conditions (*). So using again the abovementioned construction we obtain a
complex of groups C*(M) = C*(ffi) of length n.
Proposition 14. [17] Let M be a crossed ncube of groups. Then l(En(M)*) n
and there is a natural morphism of complexes NE(n)(M)* ! C*(M) which induces
isomorphisms of groups
ßi(E(n)(M)*) ~=Hi(C*(M)) , i 0 .
n ~i
Moreover ßn(E(n)(M)*) ~= \ Ker(M! M \{i}).
i=1
GENERALISED HOPF TYPE FORMULAS 19
Proof.This is obvious for n = 1. Let n = 2 and M be a crossed square respective*
*ly.
If we consider M as a crossed module of crossed modules M1 ! M0 inducing
the natural morphism of simplicial groups E(1)(M1)* !ffE(1)(M0)* satisfying the
conditions (**), then by definition E(2)(M)* = M*(ff), and by Proposition 13 and
the corresponding Remark, l(E(2)(M)*) 2, and there exists a natural morphism
of complexes NE(2)(M)* ! C*(eff) inducing an isomorphism
ßi(E(2)(M)*) ~=Hi(C*(eff)) , i 0 .
Clearly C*(eff) ~=C*(M).
Proceeding by induction, we suppose that the assertion is true for n  1 and *
*we
will show it for n.
Let us consider any crossed ncube M as a crossed module of crossed (n  1)
cubes M1 ! M0. This implies a morphism of simplicial groups E(n1)(M1)* !ff
E(n1)(M0)* satisfying the conditions (**) and a morphism of complexes C*(M1) !*
*ffi
C*(M0) satisfying the conditions (*). By definition E(n)(M)* = M*(ff), hence
Proposition 13 and its Remark imply that l(E(n)(M)*) n and there exists a
natural morphism of complexes NE(n)(M)* ~!C*(eff) inducing isomorphisms
ßi(E(n)(M)*) ~=Hi(C*(eff)), i 0.
Using the inductive hypothesis, there exist natural morphisms of complexes
0 ~00
NE(n1)(M1)* ~!C*(M1) and NE(n1)(M0)* ! C*(M0),
which induce isomorphisms
ßi(E(n1)(M1)*) ~=Hi(C*(M1)),
ßi(E(n1)(M0)*) ~=Hi(C*(M0)),
for i 0. It is easy to check that ~00eff= ffi~0 and that (~0i, ~00i) is a mo*
*rphism of
*
*0o~00
crossed modules for all i 0. Then the natural morphism of complexes C*(eff) ~*
*!
C*(ffi) = C*(M), by (2) and the five lemma, induces Hi(C*(eff)) ~=Hi(C*(M)), i *
* 0.
Therefore the morphism of complexes
(~0o~00)O~
NE(n)(M)* ! C*(M)
induces
ßi(E(n)(M)*) ~=Hi(C*(M)), i 0.
From these isomorphisms and the construction of C*(M) follows that
n ~i
ßn(E(n)(M)*) ~= \ Ker(M! M \{i}) .
i=1
20 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
5. nfold ~Cech derived functors
The ~Cech derived functors of group valued functors were introduced in [22] (*
*see
also [12] and, here, our Section 1) as an algebraic analogue of the ~Cech (co)h*
*omology
construction of open covers of topological spaces with coefficients in sheaves *
*of
abelian groups (see [29]). It is well known that the ~Cech cohomlogy of topolo*
*g
ical spaces with coefficients in sheaves is closely related to sheaf cohomology*
* of
topological spaces, in particular this relation is expressed by spectral sequen*
*ces [29].
Some applications of ~Cech derived functors to group (co)homology theory and K
theory are given in [22, 23, 24]. In this section we generalise the notion of t*
*he ~Cech
derived functors to that of the nfold ~Cech derived functors of an endofunctor*
* on the
category of groups. Based on this notion we get a new purely algebraic method f*
*or
the investigation of higher integral homology of groups from a Hopf formula poi*
*nt
of view and the further generalizations of these formulae.
Let us consider again the set = {1, . .,.n}. The subsets of are order*
*ed by
inclusion. This ordered set determines in the usual way a category Cn_. For e*
*very
pair (A, B) of subsets with A B , there is the unique morphism æAB: A ! B
in Cn_. It is easy to see that any morphism in the category Cn_, not an identi*
*ty, is
generated by æABfor all A , A 6= , B = A [ {j}, j =2A.
An ncube of groups is a functor F : Cn_! Gr , A 7! FA, æAB7! ffAB. A morphism
between ncubes F, Q : Cn_! Gr is a natural transformation ~ : F ! Q.
Warning: A crossed ncube of groups gives an ncube on forgetting structure,
but note that there is a reversal of the role of the index A. The top corner o*
*f a
crossed ncube is G, that in an ncube is F;. This is due to the fact that *
*an
ncube of groups yields a crossed ncube as a sort of generalized kernel.
Let A and consider two full subcategories of the category Cn_: CAn_is t*
*he
__
category of all subsets of containing the subset A and CAn_is the category *
*of all
subsets of not containing_the subset A. For a given ncube of groups F, and*
* A
as above, denote by FA and FA the_functors induced by the restriction of the fu*
*nctor
F to the subcategories CAn_and CAn_respectively. For a given morphism of ncube*
*s of
groups ~ : F ! Q denote by ~A : FA ! QA the natural transformation induced by
restriction of the natural transformation ~.
Examples
(a) Let (F ; R1, . .,.Rn) be a normal (n + 1)ad of groups. These data natu
rally determinesQan ncube of groups F as follows: for any A , let
FA = F OE Ri; for an inclusion A B, let ffAB: FA ! FB be the natu
i2A
ral homomorphism induced by 1F . This ncube of groups will be called the
ncube of groups induced by the normal (n+1)ad of groups, (F ; R1, . .,*
*.Rn).
GENERALISED HOPF TYPE FORMULAS 21
(b) Let (G*, d00, G) be an augmented pseudosimplicial group. A natural ncube
of groups G(n) : Cn_! Gr , n 1 is defined by the following way:
G(n)A = Gn1A for allA ,
ffAA[{j}= dn1Ak1for allA 6= , j =2A ,
where G1 = G, ffi(k) = j and ffi : ! \ A is the unique mo*
*notone
bijection.
Given an ncube of groups F. It is easy to see that there exists a natural ho*
*mo
ffA
morphism FA ! limFB for any A , A 6= .
B A
Let G be a group. An ncube of groups F will be called an npresentation of t*
*he
group G if F= G. An npresentation F of G is called free if the group FA is
free for all A 6= and called exact if the homomorphism ffA is surjective fo*
*r all
A 6= . Note that a fibrant npresentation of a group G in the sense of Brown*
*Ellis
[3] is the same as a free exact npresentation of G in our sense, for a constru*
*ction
of such, see [3].
Proposition 15. Let (G*, d00, G) be an augmented pseudosimplicial group and sup
fd00
pose that d00induces a natural isomorphism ß0(G*) ! G.
(i)Then the ncube of groups G(n), n 1, is induced by the normal (n + 1)*
*ad
of groups (Gn1, Kerdn10, . .,.Kerdn1n1) i.e.
Y
G(n)A ~=Gn1OE Ker dn1i1, A .
i2A
(ii)(G*, d00, G) is aspherical if and only if the ncube of groups G(n) is a*
*n exact
npresentation of the group G for all n 1.
Proof.(i) Straightforward from the following wellknown fact on pseudosimplicial
groups:
dnj(Ker dni) = Ker dn1ifor n > 0 , 0 i < j n .
(ii) It is wellknown that asphericity of the augmented pseudosimplicial group
(G*, d00, G) is equivalent to the simplicial exactness of (G*, d00, G), which c*
*ertainly is
equivalent to the fact that G(n) is an exact npresentation of G for all n 1.
Remark. From Proposition 15(i) follows that if (F*, d00, G) is an augmented ps*
*eu
dosimplicial group such that d00induces a natural isomorphism
fd00
ß0(F*) ! G
and Fi, i 0 are free groups, then the normal (n + 1)ad of groups
(Fn1, Kerdn10, . .,.Kerdn1n1)
satisfies the conditions of Theorem BE (see also [3], [10]). Thus, if these co*
*ndi
tions were sufficient, simplicial exactness (or asphericity) of (F*, d00, G) wo*
*uld not be
needed for getting the generalised Hopf formulas for higher homology of groups.*
* It
was this, in fact, that made us suspect that these BE conditions are not suffic*
*ient.
22 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
Given an ncube of groups F, a normal (n+1)ad of groups (F ; R1, . .,.Rn), w*
*here
F = F; and Ri= Ker ff;{i}, i 2 is called the normal (n + 1)ad of groups in*
*duced
by F. If F is an exact npresentation of the group F, then the normal (n + 1*
*)ad
of groups (F ; R1, . .,.Rn) satisfies the following condition:
Y
F OE Ri~= FA for allA ,
i2A
i.e. the ncube of groups F is induced by the normal (n+1)ad of groups (F ; R1*
*, . .,.Rn)
(see [15]).
Now let F be an npresentation of the group G. Applying C~ (see Section 1) in
the nindependent directions, this construction leads naturally to an augmented*
* n
simplicial group. Taking the diagonal of this augmented nsimplicial group giv*
*es
the augmented simplicial group (C~(n)(F)*, ff, G) called an augmented nfold C~*
*ech
complex for F, where ff = ff;: F; ! G. In case F is a free exact npresentat*
*ion of
the group G, then (C~(n)(F)*, ff, G) will be called an nfold ~Cech resolution *
*of G.
Let G, H be groups. Let F and Q be npresentations of G and H respectively
and ~ : G ! H a morphism of groups. A morphism ~ : F ! Q of ncubes will be
called an extension of the group morphism ~ if ~= ~.
Theorem 16. Let F and Q be free and exact npresentations of given groups G
and H respectively. Then any morphism of groups ~ : G ! H can be extended to a
morphism ~ : F ! Q of ncubes of groups which naturally induces a morphism e~of
simplicial groups
~C(n)(F)* ff! G
e~# # ~
~C(n)(Q)* ff! H
over ~. Furthermore, any two such extensions ~, ß : F ! Q of ~ induce simplicia*
*lly
homotopic morphisms e~, eßof simplicial groups, denoted by e~' eß.
Proof.We begin by showing the existence of a morphism of ncubes of groups ~ :
F ! Q extending the morphism of groups ~ : G ! H.
Since F is free and Q is exact, there exists a homomorphism ~\{i}: F\{i*
*}!
Q\{i}for all i 2 , such that ff\{i}~\{i}= ~ff\{i}. Suppos*
*e for some
A and for all B A, B , there exists a homomorphism ~B : FB ! QB
such that ffBC~B = ~C ffBC, C B. Then as an immediate consequence one has the
induced homomorphism __~: lim FB ! lim QB . Using again the facts that F is free
B A B A __
and Q is exact there exists a homomorphism ~A : FA ! QA such that ffA~A = ~ffA.
Clearly the constructed morphism of ncubes of groups ~ : F ! Q naturally induc*
*es
a unique morphism of augmented nsimplicial groups and applying the diagonal
gives a morphism of simplicial groups e~: ~C(n)(F)* ! ~C(n)(Q)* over the morphi*
*sm
~.
We need to prove the remaining part of the assertion first in a particular ca*
*se.
Particular Case. Let ~, ß : F ! Q be two extensions of the group morphism
~ : G ! H and l 2 . Let ~{l}= ß{l}: F{l}! Q{l}, then the respective induced
GENERALISED HOPF TYPE FORMULAS 23
morphisms of simplicial groups e~, eß: C~(n)(F)* ! C~(n)(Q)* over ~ are simplic*
*ially
homotopic.
The construction of ~C(n)directly implies that for any ncube of groups F, ~C*
*(n)(F)*
is the diagonal of a bisimplicial group F** induced by applying the ordinary_~C*
*ech
complex construction C~ to the morphism of simplicial groups C~(n1)(F{l})* !
~C(n1)(F{l})*.
By assumption the extensions ~ and ß of the group morphism ~ induce a com
mutative diagram of simplicial groups
___ ~e0 ___
C~(n1)(F{l})* !! ~C(n1)(Q{l})*
ie0
# # ,
f~00
C~(n1)(F{l})* !! ~C(n1)(Q{l})*
fi00
where e~00= fß00, which implies there are morphisms of simplicial objects of si*
*mplicial
groups _
~
F** !!_ Q**
i
# #
C~(n1)(F{l})*f~00=fi00!~C(n1)(Q{l})*
over the morphism of simplicial groups e~00= fß00.
The following lemmas will be needed.
Lemma 17. Let G**, H**be bisimplicial groups and ff, fi : G**! H**morphisms of
bisimplicial groups. Let there exist a vertical (horizontal) simplicial homotop*
*y hv(hh)
between the induced morphisms of simplicial groups ffm , fim : Gm* ! Hm*(G*m !
H*m) for all m 0, such that the following conditions hold:
dhjhvi= hvidhj (dvjhhi= hhidvj).
Then the induced morphisms of simplicial groups eff, fei: G* ! H* are simpli
cially homotopic, eff' efi, where G* and H* are the diagonal simplicial group*
*s of
G** and H** respectively.
Proof.We can construct the required homotopy in the following way:
h0i= hvishi: Gnn ! Hn+1,n+1, 0 i n.
Now we have to check the standard identities for simplicial homotopy. In fact,
dv0dh0hv0sh0=dv0hv0dh0sh0= dv0hv0= ffnn ,
dvn+1dhn+1hvnshn=dvn+1hvndhn+1shn=ædvn+1hvn= finn ,
hvj1dvishj1dhi= hvj1shj1dvidhi, i < j
dvidhihvjshj=dvihvjdhishj= v v h h v h v h ,
hjdi1sjdi1= hjsjdi1di1, i > j + 1
dvj+1dhj+1hvj+1shj+1=dvj+1hvj+1dhj+1shj+1= dvj+1hvj+1= dvj+1hvjdhj+1shj= dvj+1*
*dhj+1hvjshj.
24 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
Lemma 18. Let (C~(ff)*, ff, G), (C~(fi)*, fi, H) be augmented C~ech complexes a*
*nd
f, g : ~C(ff)* ! ~C(fi)* morphisms of simplicial groups over a given group morp*
*hism
~ : G ! H. Then f and g are simplicially homotopic, f ' g.
Proof.We only construct the simplicial homotopy and leave the checking of the
corresponding identities to the reader. We define hi: ~C(ff)n ! ~C(fi)n+1, 0 *
*i n,
by
hi(x0, . .,.xn) = (g(x0), . .,.g(xi), f(xi), . .,.f(xn))
for all (x0, . .,.xn) 2 ~C(ff)n.
Returning to the main proof, using Lemma 18, it is easy to see that there exi*
*sts a
vertical homotopy hv between the induced morphisms of simplicial groups ___~m, *
*___ßm:
Fm* ! Qm* for all m 0, such that dhjhvi= hvidhj. Applying Lemma 17 there is a
simplicial homotopy between the morphisms of simplicial groups e~, eß: ~C(n)(F)*
** !
~C(n)(Q)*.
Now we return to the general case, showing for any two extensions ~, ß : F ! Q
of a group morphism ~ : G ! H the existence of extensions ~1, . .,.~n1 : F ! Q*
* of
~ such that e~' e~1, e~1' e~2, . . . , ~gn2' g~n1, ~gn1' eßwhich, of course,*
* implies
that e~' eß. In fact, we can construct an extension ~1 : F ! Q in the following*
* way:
let ~{1}1= ~{1}and ~\{1}1= ß\{1}. We complete the construction of ~1 usin*
*g the
technique above and the facts that F is a free and Q is an exact npresentation*
* of
the groups G and H respectively.
We construct an extension ~i for all 2 i n  1 as follows: let ~{i}i= ~{*
*i}i1
and ~\i= ß\. We complete again the constructing of ~i using the ab*
*ove
technique and the facts that F is free and Q is exact.
The construction of ~i, 1 i n  1, and our already proved particular case
imply that e~' e~1, e~1' e~2, . . . , ~gn2' g~n1, ~gn1' eß.
Using this comparison theorem we make the following
Definition. Let T : Gr ! Gr be a covariant functor. Define ith nfold ~Cech de*
*rived
functor LnfoldiT : Gr ! Gr , i 0, of the functor T by choosing for each G in*
* Gr ,
a free exact npresentation F and setting
LnfoldiT (G) = ßi(T ~C(n)(F)*) ,
where (C~(n)(F)*, ff, G) is the nfold ~Cech resolution of the group G for the *
*free exact
npresentation F of G.
The nfold ~Cech complexes and hence the nfold ~Cech derived functors are cl*
*osely
related to the diagonal of the nsimplicial multinerve of crossed ncubes of gr*
*oups .
In particular, we have the following
Lemma 19. Let F be an npresentation of a group G. There is an isomorphism of
simplicial groups
C~(n)(F)* ~=E(n)(M)* ,
GENERALISED HOPF TYPE FORMULAS 25
where M is the inclusion crossed ncube of groups given by the normal (n + 1)ad
of groups (F ; R1, . .,.Rn) induced by F.
Proof.For n = 1, is done in Lemma 1 and hence for general n, both constructions
gives an isomorphism of nsimplicial groups. Applying the diagonal clearly give*
*s the
result.
The following theorem gives the nth nfold ~Cech derived functor of the func*
*tor
Zk : Gr ! Gr , k 2.
Theorem 20. Let G be a group and k 2. Then there is an isomorphism
\ Ri\ k(F )
i2
LnfoldnZk(G) ~=_________________ , n 1 ,
Dk(F ; R1, . .,.Rn)
where (F ; R1, . .,.Rn) is the normal (n + 1)ad of groups induced by some free*
* exact
npresentation F of the group G.
Proof.By Definition and Lemma 19, LnfoldnZk(G) ~= ßn(ZkE(n)(M))*, where M
is the inclusion crossed ncube of groups given by the normal (n + 1)ad of gro*
*ups
(F ; R1, . .,.Rn). Hence using Proposition 12 one has an isomorphism LnfoldnZk*
*(G) ~=
ßn(E(n)Bk(M)*). Then, by Proposition 14,
^~l,
(3) LnfoldnZk(G) ~= \ Ker (Bk(M) ! Bk(M)\{l}) .
l2
Now we set up the inductive hypothesis. Let n = 1, then
i R F j R \ (F )
L1fold1Zk(G) ~=Ker _____1____ ! ______ = _1____k____.
Dk(F ; R1) k(F ) Dk(F ; R1)
Proceeding by induction, we suppose that the result is true for n  1 and we *
*will
prove it for n. ___
Let us consider l 2 . It is easy to check that F{l}is a free exact (n  1*
*)
presentation of the free group F\{l}. Here we have to use the fact that if G*
* is a
free group, then LnfoldiT (G) = 0, i > 0 and Lnfold0T (G) ~= T (G) for any fu*
*nctor
T : Gr ! Gr . Thus, because of our inductive hypothesis,
\ Ri\ k(F )
i2\{l}
(4) L(n1)foldn1Zk(F\{l}) ~=_______________________________ = 0 .
Dk(F ; R1, . .,.Rl1, Rl+1, . .,.Rn)
Now from (3) and (4) one can easily deduce that there is the isomorphism
\ Ri\ k(F )
i2
LnfoldnZk(G) ~=_________________ .
Dk(F ; R1, . .,.Rn)
Now we are ready to improve on Theorem BE, and moreover to express by gen
eralised Hopf formulae not only the nonabelian derived functors of the functor*
* Z2,
but also the derived functors of the functors Zk, k 2.
26 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
Theorem 21. Let G be a group, F be a free exact npresentation of G and k 2.
Then
\ Ri\ k(F )
i2
LnZk(G) ~=_________________ , n 1 ,
Dk(F ; R1, . .,.Rn)
where (F ; R1, . .,.Rn) is the normal (n + 1)ad of groups induced by F.
Proof.This directly follows from Corollary 10, Proposition 15(ii) and Theorem 2*
*0.
Remark. One can prove an analogous result in a more general context of Theorem
BE and Corollary 4. In particular, for a given group G and an exact npresentat*
*ion
F of G such that
L1Zk(F;) = 0 , LrZk(FA) = 0 for r = A , r = A + 1
with A a nonempty proper subset of , there is an isomorphism
\ Ri\ k(F )
i2
LnZk(G) ~=_________________ forn 1 ,
Dk(F ; R1, . .,.Rn)
where (F ; R1, . .,.Rn) is the normal (n + 1)ad of groups induced by F.
Now we concentrate our attention on the computation of 2fold C~ech derived
functors of the functor Zk : Gr ! Gr , k 2. Using the fact that Zk, k 2 i*
*s a
right exact functor one easily shows that L2fold0Zk ~=Zk. Moreover, by Proposi*
*tion
12, Proposition 14 and Lemma 19, L2foldiZk = 0 for i 3. To take into account
Theorem 20, it only remains to compute the first 2fold ~Cech derived functor o*
*f the
functor Zk.
Lemma 22. (cf. Conduch'e [7] and also, [21]) Let
8 ~ 9
< L ! M =
M = ~0# # ~
: ;
N ! P
be a crossed square. Then
H0(C*(M)) = P OE Im ~ Im ,
H1(C*(M)) ~= M xP NOE Im ~,
H2(C*(M)) = Ker ~ \ Ker ~0,
where C*(M) is the mapping cone complex of groups
fi
L ff!M o N ! P
with ff(l) = (~(l)1, ~0(l)), fi(m, n) = ~(m) (n) for all l 2 L, (m, n) 2 M o N*
* and
~ is the natural homomorphism from L to M xP N.
GENERALISED HOPF TYPE FORMULAS 27
Proof.We only prove that H1(C*(M)) ~=M xP NOE Im ~. It is easy to check that
f : Ker fi ! M xP N, given by f(m, n) = (m1, n) for all (m, n) 2 Ker fi, is
an isomorphism and Im fff = Im ~. The other results are as easy as this part to
check.
Proposition 23. For a given group G and k 2 there are isomorphisms of groups
R1 k(F ) \ R2 k(F )
L1Zk(G) ~=L2fold1Zk(G) ~=___________________ ,
(R1 \ R2) k(F )
where (F ; R1, R2) is a normal 3ad of groups induced by some free exact 2pres*
*entation
F of G.
Proof.Begin with the first isomorphism. By the construction, C~(2)(F)* is the *
*di
agonal of a bisimplicial group F** induced by applying the_ordinary ~Cech compl*
*ex
construction C~ to the morphism of ~Cech complexes C~(F{1})* ! C~(F{1})*. Now
applying the right exact functor Zk dimensionwise, denote the resulted bisimpl*
*icial
group Zk(F**). By [27] there is a spectral sequence
E2pq=) L2foldp+qZk(G) ,
where E20q= 0, q > 0 and E2p0~=L1foldpZk(G), p 0. Hence there is an isomorph*
*ism
~= 1fold
L2fold1Zk(G) ! L1 Zk(G). Now the required isomorphism follows from [26], (*
*see
also [12]).
Again use of Proposition 12, Proposition 14 and Lemma 19 implies that
L2fold1Zk(G) ~=H1(C*(Bk(M)) ,
where M is the inclusion crossed square induced by the normal 3ad of groups
(F ; R1, R2). Since the homomorphisms
e~1: Bk(M){1}! Bk(M);
and
e~2: Bk(M){2}! Bk(M);
are injections, using Lemma 22 implies the second isomorphism.
Note that for groupabelianization functor Ab = Z2 we have the following new
interpretation of the second integral group homology
R1[F, F ] \ R2[F, F ]
H2(G) ~=L2fold1Ab(G) ~=__________________ .
(R1 \ R2)[F, F ]
It is an interesting problem to investigate and to compute the functors Lnfo*
*ldiZk
for 0 < i < n, n 3.
28 GURAM DONADZE, NICK INASSARIDZE AND TIMOTHY PORTER
6. Hopf type formulas in algebraic Ktheory
Let us recall the wellknown definition of lim(1), the first derived functor *
*of the
j
functor lim (inverse limit in the category of groups)(see e.g. [12]). Let {Aj, *
*pjj+1}j
j
be a countable inverse system of groups, then
Y
lim(1){Aj, pjj+1} = AjOE ~ ,
j j
Q
where ~ is an equivalence relation on the set Aj defined as follows: {aj} ~ {*
*a0j}
j
if there exists {hj} such that {hj}{aj}{pjj+1(h1j+1)} = {a0j}.
Let {Gj*, _jj+1}j be a countable inverse system of pseudosimplicial groups Gj*
**with
_jj+1: Gj+1*! Gj*a fibration for all j 1. Let G* = lim{Gj*, _jj+1}.
j
Theorem 24. [12] There is a short exact sequence of groups
0 ! lim(1)ßn+1(Gj*) ! ßn(G*) ! limßn(Gj*) ! 0
j j
for all n 0.
Let us define the functor Z1 : Gr ! Gr as follows: for a given group G,
Z1 (G) = limZj(G); for a given group homomorphism ~ : G ! H, Z1 (~) is the
j
group homomorphism induced by the Zj(~).
It is known from [14] (see also [12]) that the nonabelian left derived funct*
*ors
LP*Z1 of the functor Z1 : Gr ! Gr are isomorphic to Quillen's Kfunctors. Thus
using Theorem 24 we deduce that there is a short exact sequence of abelian grou*
*ps
0 ! lim(1)ßn+1(F*j(GL(R))) ! Kn+1(R) ! limßn(F*j(GL(R))) ! 0 , n 0 ,
j j
where F*(GL(R)) ! GL(R) is a free pseudosimplicial resolution of the group
GL(R) and F*j(GL(R)) = Zj(F*(GL(R))).
Now according to Corollary 10 we obtain the following
Theorem 25. Let R be a ring with unit and (F*, d00, GL(R)) be a free pseudosimp*
*li
cial resolution of the general linear group GL(R). Then there is an exact seque*
*nce
GENERALISED HOPF TYPE FORMULAS 29
of abelian groups
0 n 1
( \ Ker di1) \ j(Fn)
i2
0 ! lim(1)@_________________________A  ! Kn+1 (R) !
j Dj(Fn; Kerdn0, . .,.Kerdnn)
0 n1 1
( \ Ker di1) \ j(Fn1)
i2
! lim@ _______________________________A ! 0
j Dj(Fn1; Kerdn10, . .,.Kerdn1n1)
for n 1.
Note that using Theorem 21 and its Remark, one can express Kn+1(R) in data
coming from exact (n + 1) and npresentations of the group GL(R). We hope to
return to a more detailed analysis of this in future work.
Acknowledgements
The second author would like to thank the Royal Society, and University of Wales
Bangor for their hospitality at the `initialisation' of this paper. The author*
*s were
partially supported by INTAS grant 00 566, and in addition the third author by
INTAS 971  31961. The first and the second authors were also supported by INTAS
Georgia grant No 213, whilst the second author was supported by FNRS grant
7GEPJ065513.01.
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