1.6. Inverse conjecture over the integers 95
Example 1.6.5. Let us parameterise an element of
G{0,1}2
as (g00,g01,g10,
g11). Then
HK2(G)
is generated by elements of the form (g0,g0,g0,g0)
for g0 G≥0, (id, id,g1,g1) and (id,g1, id,g1), and (id, id, id,g2) for g0
G≥0,g1 G≥1,g2 G≥2. (This does not cover all the possible faces of
{0,
1}2,
but it is easy to see that the remaining faces are redundant.) In other
words,
HK2(G)
consists of all group elements of the form (g0,g0g1,g0g1,
g0g1g1g2), where g0 G≥0, g1,g1 G≥1, and g2 G≥2. This example is
generalised in the exercise below.
Exercise 1.6.3. Define a lower face to be a face of a discrete cube {0,
1}k
in which all the frozen coefficients are equal to 0. Let us order the lower
faces as F1,...,F2k−1 in such a way that i j whenever Fi is a subface of
Fj. Let G• be a filtered group. Show that every element of
HKk(G•)
has a
unique representation of the form
2k−1
i=0
(gi)Fi , where gi G≥k−dim(Fi) and
the product is taken from left to right (say).
Exercise 1.6.4. If G is an abelian group, show that the group
HKk(G,
d)
defined in Definition 1.6.4 agrees with the group defined at the beginning of
this section for additive groups (after transcribing the former to multiplica-
tive notation).
Exercise 1.6.5. Let G• be a filtered group. Let F be an r-dimensional face
of {0,
1}k.
Identifying F with {0,
1}r
in an obvious manner, we then obtain
a restriction homomorphism from
G{0,1}k
with
GF

G{0,1}r
. Show that the
restriction of any element of
HKk(G•)
to
G{0,1}r
then lies in
HKr(G•).
Exercise 1.6.6. Let G• be a filtered group, let k 0 and l≥1 be integers,
and let g = (gω)ω∈{0,1}k and h = (hω)ω∈{0,1}k be elements of
G{0,1}k
. Let
f = (fω)ω∈{0,1}k+l be the element of
G{0,1}k+l
defined by setting fωk,ωl for
ωk {0,
1}k,ωl
{0,
1}l
to equal gωk for ωl = (1,..., 1), and equal to
gωk hωk otherwise. Show that f
HKk+l(G•)
if and only if g
HKk(G•)
and h
HKk(G•
+l),
where G•
+l
is defined in Example 1.6.3. (Hint: Use
Exercises 1.6.3, 1.6.5.)
Exercise 1.6.7. Let G• be a filtered group, let k ≥1, and let g = (gω)ω∈{0,1}k
be an element of
G{0,1}k
. We define the derivative ∂1g
G{0,1}k−1
in the
first variable to be the tuple
(gω,1gω,1
0
)ω∈{0,1}k−1 . Show that g
HKk(G•)
if
and only if the restriction of g to {0,
1}k−1
lies in
HKk−1(G•)
and ∂1g lies
in
HKk(G• +1),
where G•
+1
is defined in Example 1.6.3.
Remark 1.6.6. The Host-Kra groups of a filtered group in fact form a cubic
complex, a concept used in topology; but we will not pursue this connection
here.
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