Genera of the Torsion-Free Polyhedra
We study the genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably homotopically equivalent. More precisely, we describe the genera of the torsion-free polyhedra of dimensions not greater than 11. In particular, we find the number of...
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Kolesnyk, P.O. 2020-02-15T07:37:08Z 2020-02-15T07:37:08Z 2013 Genera of the Torsion-Free Polyhedra / P.O. Kolesnyk // Український математичний журнал. — 2013. — Т. 65, № 10. — С. 1332–1341. — Бібліогр.: 14 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165653 515.412.3+515.143+515.146 We study the genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably homotopically equivalent. More precisely, we describe the genera of the torsion-free polyhedra of dimensions not greater than 11. In particular, we find the number of stable homotopy classes in these genera. Вивчаються роди полієдрів (скінченних клітинних комплексів), тобто класи полієдрів, yci локалiзацiї яких є стабільно гомотопічно еквівалентними. А саме, описано роди поліедрів без скруту розмірності щонайбільше 11. Зокрема, обчислено кількість стабільних гомотопічних класів у цих родах. en Інститут математики НАН України Український математичний журнал Статті Genera of the Torsion-Free Polyhedra Роди поліедрів без скруту Article published earlier |
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Genera of the Torsion-Free Polyhedra |
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Genera of the Torsion-Free Polyhedra |
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Genera of the Torsion-Free Polyhedra |
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Genera of the Torsion-Free Polyhedra |
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genera of the torsion-free polyhedra |
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Kolesnyk, P.O. |
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Kolesnyk, P.O. |
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Український математичний журнал |
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Інститут математики НАН України |
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Роди поліедрів без скруту |
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We study the genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably homotopically equivalent. More precisely, we describe the genera of the torsion-free polyhedra of dimensions not greater than 11. In particular, we find the number of stable homotopy classes in these genera.
Вивчаються роди полієдрів (скінченних клітинних комплексів), тобто класи полієдрів, yci локалiзацiї яких є стабільно гомотопічно еквівалентними. А саме, описано роди поліедрів без скруту розмірності щонайбільше 11. Зокрема, обчислено кількість стабільних гомотопічних класів у цих родах.
|
| issn |
1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/165653 |
| citation_txt |
Genera of the Torsion-Free Polyhedra / P.O. Kolesnyk // Український математичний журнал. — 2013. — Т. 65, № 10. — С. 1332–1341. — Бібліогр.: 14 назв. — англ. |
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AT kolesnykpo generaofthetorsionfreepolyhedra AT kolesnykpo rodipolíedrívbezskrutu |
| first_indexed |
2025-11-25T13:04:59Z |
| last_indexed |
2025-11-25T13:04:59Z |
| _version_ |
1850512721341054976 |
| fulltext |
UDC 515.412.3+515.143+515.146
P. O. Kolesnyk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
GENERA OF THE TORSION FREE POLYHEDRA
РОДИ ПОЛIЕДРIВ БЕЗ СКРУТУ
We study genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably
homotopically equivalent. More procisely, we describe the genera of the torsion free polyhedra of dimensions not greater
than 11. In particular, we find the number of the stable homotopy classes in these genera.
Вивчаються роди полiедрiв (скiнченних клiтинних комплексiв), тобто класи полiедрiв, усi локалiзацiї яких є стабiль-
но гомотопiчно еквiвалентними. А саме, описано роди полiедрiв без скруту розмiрностi щонайбiльше 11. Зокрема,
обчислено кiлькiсть стабiльних гомотопiчних класiв у цих родах.
1. Stable category and genera. The theory of genera of polyhedra was developed in [11] analo-
gously to the theory of genera of integral representations. This paper discovered relations between
two theories and established technique for calculation of genera of polyhedra.
The present paper contains calculations of genera of particular polyhedra. Namely, genera of tor-
sion free polyhedra with integral homologies in dimensions at most 11 are described. Such polyhedra
were described in [12, 13] (see also [14]).
In this paper, the number of stable homotopy classes of genera of those torsion free polyhedra is
found. This number can only be 1, 2 or 4.
We consider polyhedra (i.e., finite CW-complexes) as objects of the stable homotopy category
S . In particular, isomorphism always means stable homotopy equivalence. An important feature of
S is that its homomorphism groups are finitely generated [7].
Let CWm
n be the full subcategory of S consisting of (n− 1)-connected polyhedra of dimension
at most n + m. The suspension functor maps CWm
n to CWm
n+1. If n > m + 1 it is an equivalence
of categories. If n = m + 1, it is an epivalence, i.e., this functor is full, dense and conservative. In
particular, it is one-to-one on the isomorphism classes of objects. We set Sm =
⋃∞
n=1 CW
m
n . So all
objects of Sm are suspensions of the objects from CWm
m+1.
Let Zp =
{ a
b
∣∣∣a, b ∈ Z, p - b
}
, where p is a prime integer. We denote by Sp the category
which has the same objects as S , but its sets of morphisms are Hosp(X,Y ) = Hos(X,Y ) ⊗ Zp.
Actually, Hosp(X,Y ) coincides with group of stable maps between p-localizations in the sense of
Artin – Mazur – Sullivan [9]. For the sake of convenience, we denote the image in Sp of a polyhedron
X by Xp.
Definition 1.1 [11]. We say that two polyhedra X and Y are in the same genus and write
X ∼ Y if Xp ' Yp for every prime p. By G(X) we denote the genus of the polyhedron X, i.e., the
full subcategory of S consisting of all polyhedra that are in the same genus as X, and by g(X) the
number of isomorphism classes in G(X). Note that it is always finite.
For any abelian group A we denote by tors(A) its torsion part, i.e., the subgroup of all torsion
elements.
Let Λ = End(X) and Λ̄ = Λ/ nil Λ, where nil Λ is the nilpotent radical of Λ. Then Λ̄ is an order
in the semisimple algebra
∏k
i=1
Mat(ri(X),Q) for some k, where ri(X) = dimQ HosQ(Si, X),
c© P. O. KOLESNYK, 2013
1332 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
GENERA OF THE TORSION FREE POLYHEDRA 1333
according to [11] (Corollary 1.8 (3)). Then g(X) = g(Λ) = g(Λ̄) [11], where g(Λ) is the number of
isomorphism classes of Λ-modules M such that Mp ' Λp for all prime p. In particular, g(Λ) < ∞
by the Jordan – Zassenhaus theorem [3] (Theorem 24.1).
2. Calculations
For calculation of g(X) for a particular polyhedron X, the following facts are used.
Proposition 2.1 [11]. Let Λ be an order in
∏k
i=1
Mat(ri,Q) for some k, Γ be a maximal order
containing Λ and Λ ⊇ mΓ for some integer m > 1. Then g(Λ) equals the number of cosets
Im γ\(Γ/mΓ)×/(Λ/mΓ)×,
where γ is the natural map Γ× → (Γ/mΓ)×.
Applied to polyhedra, it gives the following result.
Theorem 2.1 [11]. Let X be a polyhedron, B =
∨k
i=1 riS
ni with different n1, n2, . . . , nk and
some k. Suppose that there are maps X
β−→ B
α−→ X such that αβ ≡ m1X mod tors(X) and
βα ≡ m1B mod tors(B) for some integer m > 1. Then g(X) = 1 if m = 2 and g(X) ≤
(
ϕ(m)/2
)k
if m > 2.
In the following examples, definitions and results from [6] (Section 3) and [11] (Sections 1, 2)
are used. In particular, we denote by a the ath multiple of a generator of the group πn(Sn) ' Z, by
η the nonzero element of πSn (Sn−1) ' Z/2, by η2 the nonzero element of πSn (Sn−2) ' Z/2 and by
ν the generator of πSn (Sn−3) ' Z/24. We also denote by Z ×m Z the subring of Z × Z consisting
of all pairs (α, β) with α ≡ β (mod m). Note that Z×m Z ⊇ m(Z× Z), so g(Z×m Z) equals the
number of double cosets
{±1} × {±1}\Z×m × Z×m/Z×m
under the diagonal embedding of Z×m into Z×m × Z×m. It easily gives g(Z×m Z) = ϕ(m)/2.
We consider the torsion free polyhedra from subcategories S 4 and S 5 (see Section 1), i. e.,
the polyhedra X with torsion free homology groups Hi(X) for all i. Recall that these are just the
cases when there are only finitely many isomorphisms classes of torsion free polyhedra. From [12],
it is known that the torsion free polyhedra from subcategory S 4 arise from the following cofibration
sequences:
S8 f→ S5 → A(v)→ S9,
where f = v;
S7 ∨ S8 f→ S5 → A(η2v)→ S8 ∨ S9,
where f = (η2, vν);
S8 f→ S5 ∨ S7 → A(vη)→ S9,
with f =
(
vν
η
)
;
S6 ∨ S8 f→ S5 → A(ηv)→ S7 ∨ S9,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1334 P. O. KOLESNYK
with f = (η, vν);
S8 f→ S5 ∨ S6 → A(vη2)→ S9,
where f =
(
vν
η2
)
;
S7 ∨ S8 f→ S5 ∨ S7 → A(η2vη)→ S8 ∨ S9,
where f =
(
η2 vν
0 η
)
;
S7 ∨ S8 → S5 ∨ S6 → A(η2vη2)→ S8 ∨ S9,
where f =
(
η2 vν
0 η2
)
;
S6 ∨ S8 → S5 ∨ S6 → A(ηvη2)→ S7 ∨ S9,
with f =
(
η νv
0 η2
)
;
S6 ∨ S8 → S5 ∨ S7 → A(ηvη)→ S7 ∨ S9,
where f =
(
η νv
0 η
)
.
Here 1 ≤ v ≤ 6, except of the cases A(v), when 1 ≤ v ≤ 12, and A(ηvη), when 1 ≤ v ≤ 3.
We provide calculations only for A(v), A(η2v) and A(η2vη). The rest of the cases is treated in
the analogous way with the similar results.
Consider the polyhedron A(v). It follows from [6] (Theorem 2.4) that modulo the nilpotent
radical, End
(
A(v)
)
is isomorphic to the ring of pairs (α, β), where α, β ∈ Z and αvν = βvν, that
is α ≡ β (mod m), where m = 24/d and d = gcd(v, 24). It is the ring Z×m Z. Therefore
g
(
A(v)
)
=
4 if d = 1,
2 if d = 2 or d = 3,
1 if d > 3.
Actually, all atoms A(v) with fixed gcd(v, 24) are in the same genus [11].
For the polyhedron A(η2vη) the cofibration sequence is
S7 ∨ S8 f→ S5 ∨ S7 → A(η2vη)→ S8 ∨ S9
where
f =
(
η2 vν
0 η
)
and ν is the generator of πS8 (S5) ' Z/24.
To simplify the calculations of g(A), we can use the Theorem 2.2 from [6] (Section 3). It implies
that modulo the nilpotent radical, End
(
A(η2vη)
)
is isomorphic to the ring of pairs (α, β), where
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
GENERA OF THE TORSION FREE POLYHEDRA 1335
α : S7 ∨ S8 → S7 ∨ S8 and β : S5 ∨ S7 → S5 ∨ S7 such that fα = βf, modulo the pairs (γf, fγ′),
where both γ and γ′ are the maps S5 ∨ S7 → S7 ∨ S8. Here
α =
(
a bη
0 c
)
, β =
(
x yη
0 z
)
for some integers a, b, c, x, y, z. The equation fα = βf turns into
xη2 = aη2, xvν + 12yν = 12bν + vcν, zη = cη,
that is
x ≡ a(mod 2), xv ≡ cv(mod 12), z ≡ c(mod 2),
because η3 = 12ν. Both γ and γ′ are of the form
(
0 t
0 0
)
, so the products γf, fγ′ are of the form(
0 tη
0 0
)
and
(
0 tη2
0 0
)
for some t ∈ Z. It kills bη in α and yη2 in β, and we obtain the subring
Λ of Γ = Z4 consisting of the quadruples (a, c, x, z) such that
a ≡ c ≡ x ≡ z(mod 2), c ≡ x(modm),
where m = 12/gcd(v, 12), if m 6= 3. If m = 3, then we have
a ≡ x(mod 2), c ≡ z(mod 2), c ≡ x(mod 3).
If m 6= 3, then the cosets from Proposition 2.1
Im γ\(Γ/mΓ)×/(Λ/mΓ)×
which describe g(A) become
U\C×2 × C
×
m × C×m × C×2 /V,
where U = {±1} × {±1} × {±1} × {±1}, Cm = (Z/m)× and
V =
{
(a, c, x, z) | a ≡ c ≡ x ≡ z(mod 2), c ≡ x(modm)
}
.
In case of m = 3 we have Λ ⊃ 6Γ, therefore g(A) = 1. So, we always set g(A) = ϕ(m)/2, which
equals 1 if m ≤ 6 and 2 if m = 12. Hence
g(A(η2vη)) =
2 if v = 1 or v = 5,
1 otherwise.
For the polyhedron A(η2v) the cofibration sequence is
S7 ∨ S8 f→ S5 → A(η2v)→ S8 ∨ S9,
where f =
(
η2 vν
)
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1336 P. O. KOLESNYK
As it was done in the previous case, we shall use Theorem 2.2 from [6] (Section 3). Here modulo
the nilpotent radical, End(A(η2v)) is isomorphic to the ring of pairs (α, β) with α : S7∨S8 → S7∨S8
and β : S5 → S5 such that fα = βf since there are no nonzero maps S5 → S7 ∨ S8. Here
α =
(
a bη
0 c
)
, β = x
for some integers a, b, c, x. The equation fα = βf becomes
xη2 = aη2,
xvν = 12bν + cvν,
that is
x ≡ a(mod 2),
xv ≡ cv(mod 12).
Modulo the nilpotent radical, we obtain the subring Λ of Γ = Z3 consisting of the triples (a, c, x)
such that
a ≡ c ≡ x(mod 2), c ≡ x(modm),
where m = 12/gcd(v, 12) (case m = 3 is treated analogously to the previous example). Thus
Γ×\
∏
p|m
Γ×p /
∏
p|m
Λ×p ' U\C×2 × C
×
m × C×m/V,
where U = {±1} × {±1} × {±1}, Cm = (Z/m)× and
V = {(a, c, x) | a ≡ c ≡ x(mod 2), c ≡ x(modm)}.
It gives g(A) = ϕ(m)/2, which equals 1 if m ≤ 6 and 2 if m = 12. Hence
g(A(η2v)) =
2 if v = 1 or v = 5,
1 otherwise.
As the result we obtain
g(A(v)) =
4 if d = 1,
2 if d = 2 or d = 3,
1 if d > 3,
where d = gcd(v, 24), and
g(A) =
2 if v = 1 or v = 5,
1 otherwise,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
GENERA OF THE TORSION FREE POLYHEDRA 1337
in all other cases of the torsion free polyhedra A from S 4.
Now we treat the Baues – Drozd A-atoms from S 5 [6] (Section 5). In all cases v, w ∈ {1, 2, 3, 4, 5, 6}.
These atoms are defined by the following cofibration sequences:
S9 ∨ S10 f→ S6 ∨ S7 → A(vη2w)→ S10 ∨ S11,
where f =
(
vν1 0
η2 wν2
)
and ν1, ν2 are the generators of πS9 (S6) ' Z/24 and πS10(S
7) ' Z/24
respectively;
S8 ∨ S9 ∨ S10 f→ S6 ∨ S7 ∨ S9 → A(η2vη2wη)→ S9 ∨ S10 ∨ S11,
with f =
η2 vν1 0
0 η2 wν2
0 0 η
;
S9 ∨ S10 f→ S6 ∨ S7 ∨ S8 → A(vη2wη2)→ S10 ∨ S11,
where f =
vν1 0
η2 wν2
0 η2
;
S9 ∨ S10 f→ S6 ∨ S7 ∨ S9 → A(vη2wη)→ S10 ∨ S11,
where f =
vν1 0
η2 wν2
0 η
;
S8 ∨ S9 ∨ S10 → S6 ∨ S7 ∨ S8 → A(η2vη2wη2)→ S9 ∨ S10 ∨ S11,
with f =
η2 vν1 0
0 η2 wν2
0 0 η2
;
S7 ∨ S9 ∨ S10 → S6 ∨ S7 ∨ S9 → A(ηvη2wη)→ S8 ∨ S10 ∨ S11,
with f =
η vν1 0
0 η2 wν2
0 0 η
;
S7 ∨ S9 ∨ S10 → S6 ∨ S7 → A(ηvη2w)→ S8 ∨ S10 ∨ S11,
where f =
(
η vν1 0
0 η2 wν2
)
;
S8 ∨ S9 ∨ S10 → S6 ∨ S7 → A(η2vη2w)→ S9 ∨ S10 ∨ S11,
where f =
(
η2 vν1 0
0 η2 wν2
)
;
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1338 P. O. KOLESNYK
S7 ∨ S9 ∨ S10 → S6 ∨ S7 ∨ S8 → A(ηvη2wη2)→ S8 ∨ S10 ∨ S11,
with f =
η vν1 0
0 η2 wν2
0 0 η2
.
We provide calculations only for A(vη2w), A(ηvη2w) and A(η2vη2wη). Other cases are treated
in the analogous way giving the similar results.
For A(vη2w) the cofibration sequence is
S9 ∨ S10 f→ S6 ∨ S7 → A(vη2w)→ S10 ∨ S11,
where f =
(
vν1 0
η2 wν2
)
. Analogously to the previous cases, we apply the Theorem 2.2 from [6]
(Section 3). Here α : S9 ∨ S10 → S9 ∨ S10 and β : S6 ∨ S7 → S6 ∨ S7. γ and γ′ are zero maps and
α =
(
a bη
0 c
)
, β =
(
x yη
0 z
)
for some integers a, b, c, x, y, z. The equation fα = βf implies
cw ≡ zw(mod 12),
xv ≡ av(mod 12),
z ≡ a(mod 2),
and we obtain the subring Λ of Γ = Z4 consisting of the quadruples (a, c, x, z) such that
a ≡ z(mod 2), a ≡ x(modm), c ≡ z(modm′),
where m = gcd(v, 12) and m′ = gcd(w, 12), if m,m′ 6= 3. Suppose that both m,m′ are even. Then
we have
a ≡ x(modm), c ≡ z(modm′),
so that the cosets are defined by the subring Λ× ⊃
(
a ≡ (m), c ≡ µ(m′), x ≡ (m), z ≡ µ(m′)
)
and
by (Z/m)× × (Z/m)×. Therefore g(A) = ϕ(m)/2× ϕ(m′)/2.
Assume now that m′ = 3 and m is even. Then
a ≡ x(modm), c ≡ z(mod 3), a ≡ z(mod 2),
so that the cosets are defined by (a ≡ 1(m), c ≡ 1(6), x ≡ 1(m), z ≡ 1(3)), wherefrom g(A) =
= ϕ(m)/2× ϕ(m′)/2. Hence
g(A(vη2w)) =
1 if v, w ∈ {2, 3, 4, 6},
2 if v = 1 or v = 5, w ∈ {2, 3, 4, 6},
2 if w = 1 or w = 5, v ∈ {2, 3, 4, 6},
4 in all other cases.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
GENERA OF THE TORSION FREE POLYHEDRA 1339
The polyhedron A(ηvη2w) comes from the cofibration sequence
S7 ∨ S9 ∨ S10 f→ S6 ∨ S7 → A(ηvη2w)→ S8 ∨ S10 ∨ S11,
where f =
(
η vν1 0
0 η2 wν2
)
.
In this case α : S7 ∨ S9 ∨ S10 → S7 ∨ S9 ∨ S10 and β : S6 ∨ S7 → S6 ∨ S7 so that
α =
a bη2 wαν2
0 c dη
0 0 g
, β =
(
x yη
0 z
)
for some integers a, b, c, d, g, wα, x, y, z. The equation fα = βf gives
cv ≡ xv(mod 12),
gw ≡ zw(mod 12),
z ≡ c(mod 2),
a ≡ x(mod 2).
Here γ and γ′ are of the form
0 t
0 0
0 0
, so the products γf, fγ′ are of the form
0 tη2 twν2
0 0 0
0 0 0
and
(
0 tη
0 0
)
for some t ∈ Z. It kills bη2 and wαν2 in α and yη in β
and we get the subring Λ of Γ = Z6 consisting of the elements (a, c, d, g, x, z) such that
a ≡ c ≡ x ≡ z ≡ g(mod 2),
c ≡ x(modm), g ≡ z(modm′),
where m = gcd(v, 12) and m′ = gcd(w, 12).
Applying the same considerations as in the previous case, we obtain that g(A) = ϕ(m)/2 ×
× ϕ(m′)/2. Hence
g(A(ηvη2w)) =
1 if v, w ∈ {2, 3, 4, 6},
2 if v = 1 or v = 5, w ∈ {2, 3, 4, 6},
2 if w = 1 or w = 5, v ∈ {2, 3, 4, 6},
4 in all other cases .
The polyhedron A(η2vη2wη) appears in the cofibration sequence
S8 ∨ S9 ∨ S10 f→ S6 ∨ S7 ∨ S9 → A(η2vη2wη)→ S9 ∨ S10 ∨ S11,
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1340 P. O. KOLESNYK
where f =
η2 vν1 0
0 η2 wν2
0 0 η
.
In this case α : S8 ∨ S9 ∨ S10 → S8 ∨ S9 ∨ S10 and β : S6 ∨ S7 ∨ S9 → S6 ∨ S7 ∨ S9 so that
α =
a bη cη2
0 d fη
0 0 g
, β =
x yη vβν1
0 z pη2
0 0 q
for some integers a, b, c, d, f, g, vβ, x, y, z, p, q. The equation fα = βf implies
dv ≡ xv(mod 12), gw ≡ zw(mod 12), z ≡ d(mod 2),
a ≡ x(mod 2), g ≡ q(mod 2),
and γ and γ′ are of the form
0 0 tη
0 0 u
0 0 0
, so the products γf, fγ′ are of the form
0 0 tη2
0 0 uη
0 0 0
and
0 0 tη3 + uvν1
0 0 uη2
0 0 0
for some t, u ∈ Z.
It kills cη2 and fη in α and pη2 and vβν1 in β and we get the subring Λ of Γ = Z8 consisting of
the elements (a, b, d, g, x, y, z, q) such that
a ≡ x ≡ d ≡ z ≡ g ≡ q(mod 2),
d ≡ x(modm), g ≡ z(modm′),
where m = gcd(v, 12) and m′ = gcd(w, 12).
Again, as in the previous case, we obtain that g(A) = ϕ(m)/2× ϕ(m′)/2 and therefore
g
(
A(η2vη2wη)
)
=
1 if v, w ∈ {2, 3, 4, 6},
2 if v = 1 or v = 5, w ∈ {2, 3, 4, 6},
2 if w = 1 or w = 5, v ∈ {2, 3, 4, 6},
4 in all other cases.
Acknowledgements
The author is deeply grateful to his research advisor Yurii Drozd.
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Received 10.01.13,
after revision — 30.06.13
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