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.