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
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