1.6. Inverse conjecture over the integers 97

is polynomial from H• to G•, and h ∈ H≥i for some i ≥ 0, then Δhφ is

polynomial from H• to G•

+i.

But this is easily seen from Exercise 1.6.7.

Now we establish the “if” direction. We need to show that φ maps

HKk(H•)

to

HKk(G•)

for each k. We establish this by induction on k. The

case k = 0 is trivial, so suppose that k ≥ 1 and that the claim has already

been established for all smaller values of k.

Let h ∈

HKk(H•).

We split

H{0,1}k

as

H{0,1}k−1

×

H{0,1}k−1

. From Ex-

ercise 1.6.7 we see that we can write h = (h0,h1h0) where h0 ∈

HKk−1(H•)

and h1 ∈

HKk−1(H• +1),

thus φ(h) = (φ(h0),φ(h1h0)) (extending φ to act

on

H{0,1}k−1

or

H{0,1}k

in the obvious manner). By induction hypothe-

sis, φ(h0) ∈

HKk−1(G•),

so by Exercise 1.6.7, it suﬃces to show that

φ(h1h0)φ−1(h0)

∈

HKk−1(G• +1).

By telescoping series, it suﬃces to establish this when h1 = hF for some

face F of some dimension r in {0,

1}k−1

and some h ∈ H≥k−r, as these

elements generate

HKk−1(H•

+1).

But then

φ(h1h0)φ−1(h0)

vanishes outside

of F and is equal to Δh1 φ(h0) on F , so by Exercise 1.6.6 it will suﬃce to

show that Δh1 φ(h0) ∈

HKr(G•

+k−r),

where h0 is h0 restricted to F (which

one then identifies with {0,

1}r).

But by the induction hypothesis, Δh1 φ

maps

HKr(H•)

to

HKr(H•

+k−r),

and the claim then follows from Exercise

1.6.5.

Exercise 1.6.8. Let i1,...,ik ≥ 0 be integers. If G• is a filtered group,

define

HK(i1,...,ik)(G•)

to be the subgroup of

G{0,1}k

generated by the ele-

ments gF , where F ranges over all faces of {0,

1}k

and g ∈ G≥ij1

+···+ijr

,

where 1 ≤ j1 · · · jr ≤ k are the coordinates of F that are frozen.

This generalises the Host-Kra groups

HKk(G•),

which correspond to the

case i1 = · · · = ik = 1. Show that if φ is a polynomial map from H• to G•,

then φ maps

HK(i1,...,ik)(H•)

to

HK(i1,...,ik)(G•).

Exercise 1.6.9. Suppose that φ: H → G is a non-classical polynomial of

degree ≤ d from one additive group to another. Show that φ is a polynomial

map from (H, ≤ m) to (G, ≤ dm) for every m ≥ 1. Conclude, in particular,

that the composition of a non-classical polynomial of degree ≤ d and a non-

classical polynomial of degree ≤ d is a non-classical polynomial of degree

≤ dd .

Exercise 1.6.10. Let φ1 : H → G1, φ2 : H → G2 be non-classical polyno-

mials of degrees ≤ d1, ≤ d2, respectively, between additive groups H, G1,G2,

and let B : G1 × G2 → G be a bihomomorphism to another additive group

(i.e. B is a homomorphism in each variable separately). Show that B(φ1,φ2):

H → G is a non-classical polynomial of degree ≤ d1 + d2.

1.6.2. Nilsequences. We now specialise the above theory of polynomial

maps φ: H → G to the case when H is just the integers Z = (Z, ≤ 1) (viewed