Atoms in the p-localization of Stable Homotopy Category
We study p-localizations, where p is an odd prime, of the full subcategories \( {\mathcal{S}}^n \) of stable homotopy category formed by CW-complexes with cells in n successive dimensions. Using the technique of triangulated categories and matrix problems, we classify the atoms (indecomposable obj...
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| author | Drozd, Yu. A. Kolesnyk, P. O. Дрозд, Ю. А. Колесник, П. О. |
| author_facet | Drozd, Yu. A. Kolesnyk, P. O. Дрозд, Ю. А. Колесник, П. О. |
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| datestamp_date | 2019-12-05T10:25:15Z |
| description | We study p-localizations, where p is an odd prime, of the full subcategories \( {\mathcal{S}}^n \) of stable homotopy category formed by CW-complexes with cells in n successive dimensions. Using the technique of triangulated categories and matrix problems, we classify the atoms (indecomposable objects) in \( {\mathcal{S}}_p^n \) for n ≤ 4(p − 1) and show that, for n > 4(p − 1), this classification is wild in a sense of the representation theory. |
| first_indexed | 2026-03-24T02:19:37Z |
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UDC 512.5
Yu. A. Drozd, P. O. Kolesnyk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY
АТОМИ В p-ЛОКАЛIЗАЦIЇ СТАБIЛЬНОЇ ГОМОТОПIЧНОЇ КАТЕГОРIЇ
We study p-localizations, where p is an odd prime, of the full subcategories Sn of stable homotopy category consisting
of CW-complexes having cells in n successive dimensions. Using the technique of triangulated categories and matrix
problems, we classify atoms (indecomposable objects) in Sn
p for n ≤ 4(p − 1) and show that, for n > 4(p − 1), this
classification is wild in the sense of the representation theory.
Вивчаються p-локалiзацiї (де p — непарне просте число) повних пiдкатегорiй Sn стабiльної гомотопiчної категорiї,
що складається з CW-комплексiв iз клiтинами в n послiдовних розмiрностях. Застосовуючи технiку триангульованих
категорiй та матричнi задачi, ми наводимо класифiкацiю атомiв (нерозкладних об’єктiв) у Sn
p для n ≤ 4(p − 1) i
показуємо, що для n > 4(p− 1) ця класифiкацiя є дикою у сенсi теорiї зображень.
Introduction. Classification of homotopy types of polyhedra (finite CW-complexes) is an old prob-
lem. It is well-known that it becomes essentially simpler if we consider the stable situation, i.e.,
identify two polyhedra having homotopy equivalent (iterated) suspensions. It leads to the notion of
stable homotopy category and stable homotopy equivalence. Such a classification has been made for
polyhedra of low dimensions by several authors; a good survey of these results is the paper of Baues
[2]. Unfortunately, it cannot be done for higher dimensions, since the problem becomes extremely
complicated. Actually, it results in “wild problems” of the representation theory, i.e., problems con-
taining classification of representations of all finitely generated algebras over a field (cf. [3, 10, 11];
for generalities about wild problems see the survey [9]).
In the survey [10] the first author proposed a new approach to the stable homotopy classification
which seems more “algebraic” and simpler for calculations. It is based on the triangulated structure of
the stable homotopy category and uses the technique of “matrix problems”, more exactly, bimodule
categories in the sense of [9]. In particular, it gave simplified proofs of the results of [3 – 5]. In [11]
this technique gave new results on classification of polyhedra with torsion free homologies.
The main difficulties in the stable homotopy classification are related to the 2-components of
homotopy groups. That is why it is natural to study p-local polyhedra, where p is an odd prime;
then we only use the p-parts of homotopy groups. In this paper we use the technique of [10, 11]
to classify p-local polyhedra that only have cells in n successive dimensions for n ≤ 4(p − 1).
Analogous results have been obtained by Henn [13], who used a different approach. Our description
seems more straightforward and more visual. It gives explicit construction of polyhedra by successive
attaching simpler polyhedra to each other. We also show that for n > 4(p−1) the stable classification
of p-local polyhedra becomes a wild problem, so the obtained results are in some sense closing.
Section 1 covers the main notions from the stable homotopy theory, bimodule categories and
their relations. In Section 2 we calculate morphisms between Moore polyhedra and their products. In
Section 3 we describe polyhedra in the case n = 2p−1. This classification happens to be “essentially
finite” in the sense that there is an upper bound for the number of cells in indecomposable polyhedra
(atoms); actually, atoms have at most 4 cells. Section 4 is the main one. Here we describe polyhedra
for 2p ≤ n ≤ 4(p − 1). The result is presented in terms of strings and bands, which is usual
c© YU. A. DROZD, P. O. KOLESNYK, 2014
458 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 459
in the modern representation theory. String and band polyhedra are defined by some combinatorial
invariant (a word) and, in band case, an irreducible polynomial over the residue field Z/p. In the
representation theory such description is said to be tame. Finally, in Section 5 we prove that the
classification becomes wild if n > 4(p− 1).
The description obtained by matrix methods is local, just as that of [13]. Using the results of [12]
we also obtain a global description of p-primary polyhedra. Fortunately, it almost coincides with the
local one, except rare special cases, when one local object gives rise to (p− 1)/2 global ones.
The first author expresses his thanks to H.-J. Baues, who introduced him into the world of
algebraic topology and was his coauthor in several first papers on this topic.
1. Stable homotopy category and bimodule categories. We use basic definitions and facts
concerning stable homotopy from [8]. We denote by S the stable homotopy category of polyhedra,
i.e., finite CW-complexes. It is an additive category and the morphism groups in it are Hos(X,Y ) =
= lim−→k
Hot
(
X[k], Y [k]
)
, where X[k] denotes the k-fold suspension of X and Hot(X,Y ) denotes
the set of homotopy classes of continuous maps X → Y. Note that the direct sum in this category is
the wedge (bouquet, or one-point gluing) X∨Y and the natural map Hos(X,Y )→ Hos
(
X[k], Y [k]
)
is an isomorphism. In what follows, we always deal with polyhedra as the objects of this category.
In particular, isomorphism means stable homotopy equivalence. Note that all groups Hos(X,Y ) are
finitely generated and the stable homotopy groups πSn (X) = Hos(Sn, X) are torsion if n > dimX.
It is convenient to formally add to S the “negative shifts” X[−k] k ∈ N of polyhedra with the natural
sets of morphisms, so that X[k][l] ' X[k + l] and Hos(X[k], Y [k]) ' Hos(X,Y ) for all k ∈ Z.
Then S becomes a triangulated category, where the suspension plays role of the shift and the exact
triangles are cofibre sequences X → Y → Z → X[1] (in S they are the same as fibre sequences).
From now on we consider S with these additional objects. Actually, the category obtained in this way
is equivalent to the category of finite S-spectra [8, 15].
We denote by Sn the full subcategory of S whose objects are the shifts X[k], k ∈ Z, of polyhedra
only having cells in at most n successive dimensions, or, the same, (m − 1)-connected and of
dimension at most n + m for some m. The Freudenthal theorem [8] (Theorem 1.21) implies that
every object of Sn is a shift (iterated suspension) of an n-connected polyhedron of dimension at most
2n− 1. We denote the full subcategory of Sn consisting of such polyhedra by S
n
. Moreover, if two
such polyhedra are isomorphic in S, they are homotopy equivalent. Following Baues [2], we call an
object from Sn an atom if it belongs to S
n
, does not belong to Sn−1 and is indecomposable (into a
wedge of non-contractible polyhedra).
Recall that the p-localization of an additive category C is the category Cp such that ObCp = ObC
and HomCp(A,B) = Zp ⊗ HomC(A,B), where Zp ⊂ Q is the subring
{ a
b
∣∣∣ a, b ∈ Z, p - b} .
We consider the localized categories Sp and Snp and denote their groups of morphisms X → Y
by Hosp(X,Y ). Actually, Sp coincides with the stable homotopy category of finite p-local CW-
complexes in the sense of [14]. Every such space can be considered an image in Sp of a p-primary
polyhedron, i.e., such polyhedron X that the map pk1X for some k can be factored through a wedge
of spheres [8].
To study the categories Snp we use the technique of bimodule categories, like in [11]. We recall
the corresponding notions.
Definition 1.1 (cf. [9], Section 4). Let A and B be additive categories, M be an A-B-bimodule,
i.e., a biadditive functor Aop × B → Ab (the category of abelian groups). The bimodule category
E(M) (or the category of elements of M) is defined as follows:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
460 YU. A. DROZD, P. O. KOLESNYK
ObE(M) =
⋃
A∈ObA
B∈ObB
M(A,B).
If u ∈M(A,B), v ∈M(A′, B′), then
HomE(M)(u, v) =
{
(f, g) | f : A→ A′, g : B → B′, gu = vf
}
(
both these elements are from M(A,B′)
)
. E(M) is also an additive category. Note that we only
consider bipartite bimodules in the sense of [9].
Usually we choose a set of additive generators of A and B, i.e., sets {A1, A2, . . . , As} ⊂ ObA
and {B1, B2, . . . , Br} ⊂ ObB such that every object from A (respectively, from B) is isomorphic
to a direct sum
⊕s
j=1 kjAj (respectively,
⊕r
i=1 liBi). Then an object of E(M) can be presented as a
block matrix F = (Fij), where Fij is a matrix of size li × kj with coefficients from M(Aj , Bi). If
we present morphisms in the analogous matrix form, the action of morphisms on elements from M
is presented by the usual matrix multiplication.
We use the following localized version of [11] (Theorem 2.2).
Theorem 1.1. Let n ≤ m < 2n − 1. Denote by A (respectively, by B) the full subcategory
of Sp consisting of (m − 1)-connected polyhedra of dimension at most 2n − 2 (respectively, of
(n − 1)-connected polyhedra of dimension at most m). Consider the A-B-bimodule M such that
M(A,B) = Hosp(A,B). Let I be the ideal of the category E(M) consisting of all morphisms
(α, β) : f → f ′ such that α factors through f and β factors through f ′. Let also J be the ideal of S
n
p
consisting of all maps f : X → Y such that f factors both through an object from A[1] and through
an object from B. The map f 7→ Cf (the cone of f ) induces an equivalence E(M)/I ' S
n
p/J.
Moreover, J2 = 0, hence the isomorphism classes of the categories S
n
p and S
n
p/J are the same.
Note also that all groups J(X,Y ) are finite [12] (Corollary 1.10).
Finally, recall that, for k < l < k + 2p(p − 1) − 1, the only nontrivial p-components of the
stable homotopy groups Hos(Sl, Sk) are Hosp(S
k+qs , Sk) = Z/p, where 1 ≤ s < p and qs =
= 2s(p− 1)− 1 [16].
2. Moore polyhedra. The only atoms in S2p are Moore atoms Mk (k ∈ N) which are cones of
the maps S2 pk−→ S2. We denote their d-dimensional suspensions Mk[d − 3] by Md
k and call them
Moore polyhedra. For unification, we denote Sd by Md
0 . We need to know the morphism groups
Mdr
kl = Hosp(M
r
l ,M
d
k ). We always suppose that d − 1 ≤ r < d + 2p − 1. Obviously, Mdd
00 = Zp,
Md,d+2p−3
00 = Z/p andMdr
00 = 0 if r /∈ {d, d+ 2p− 3} . If k > 0, from the cofibre sequences
Sd−1
pk−→ Sd−1 →Md
k → Sd
pk−→ Sd (Edk)
one easily obtains thatMdr
0k =Mdr
k0 = 0, except the cases
Md,d−1
k0 'Mdd
0k ' Z/pk,
Md,d+2p−3
k0 'Md,d+2p−3
0k 'Md,d+2p−4
k0 'Md,d+2p−2
0k ' Z/p.
The values ofMdr
kl for k, l ∈ N, d− 1 ≤ r < d+ 2p− 1 can be obtained if we apply Hosp(Mr
l , )
to the cofibre sequences (Edk). It gives exact sequences
Md−1,r
0l
pk−→Md−1,r
0l →Mdr
kl →Mdr
0l
pk−→Mdr
0l ,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 461
whence we get
Mdr
kl =
Z/pmin(k,l) if r ∈ {d− 1, d} ,
Z/p if r ∈ {d+ 2p− 2, d+ 2p− 4} ,
Z/p⊕ Z/p if r = d+ 2p− 3,
0 in other cases.
(2.1)
The only nontrivial value here is for r = d+ 2p− 3: we need to know that the exact sequence
0→ Z/p α−→Md,d+2p−3
kl
β−→ Z/p→ 0 (2.2)
splits. It splits indeed for k = 1 since the middle term is a module over Mdd
11 = Z/p. If k > 1,
suppose that the sequence forMd,d+2p−3
k−1,l splits. The commutative diagram
Sd−1
pk−−−−→ Sd−1 −−−−→ Md
k −−−−→ Sd
pk−−−−→ Sd
p
y 1
y y p
y 1
y
Sd−1
pk−1
−−−−→ Sd−1 −−−−→ Md
k−1 −−−−→ Sd
pk−1
−−−−→ Sd
(2.3)
induces the commutative diagram
0 −−−−→ Z/p −−−−→ Md,d+2p−3
kl −−−−→ Z/p −−−−→ 0
1
y y 0
y
0 −−−−→ Z/p −−−−→ Md,d+2p−3
k−1,l −−−−→ Z/p −−−−→ 0.
Since the second row splits, the first one splits as well. Therefore, the sequence (2.2) splits for all
values of k and l.
Definition 2.1. We fix generators of the groupsMdr
kl and denote, for r = d+ 2p− 3,
by αd
∗
∗
kl (k, l ∈ N) the generator ofMd+1,r+1
kl which is in the image of the map α from (2.2);
by αdkl (k, l ∈ N ∪ {0}) the generator ofMdr
kl which is not in Imα;
by αd
∗
kl (k ∈ N ∪ {0}, l ∈ N) the generator ofMd,r+1
kl ;
by αd∗kl (k ∈ N, l ∈ N ∪ {0}) the generator ofMd+1,r
kl ;
by γdkl (k, l ∈ N ∪ {0}) the generator ofMdd
kl ;
by γd∗kl (k ∈ N, l ∈ N ∪ {0}) the generator ofMd+1,d
kl .
Note that all these morphisms are actually induced by maps Sr → Sd. Using diagrams of the
sort (2.3), one easily verifies that these generators can be so chosen that
α
d∗∗
kl γ
r+1
ll′ =
α
d∗∗
kl′ if l ≤ l′,
0 if l > l′,
αd
∗
kl γ
r+1
ll′ =
α
d∗
kl′ if l ≤ l′,
0 if l > l′,
αdklγ
r
ll′ =
α
d
kl′ if l ≥ l′ or l = 0,
0 if 0 < l < l′,
αd∗kl γ
r
ll′ =
α
d∗
kl′ if l ≥ l′ or l = 0,
0 if 0 < l < l′,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
462 YU. A. DROZD, P. O. KOLESNYK
αd
∗
kl γ
r∗
lk′ = αdkk′ , α
d∗∗
kl γ
r∗
lk′ = αd∗kk′ ,
(2.4)
γd+1
k′k α
d∗∗
kl =
{
α
d∗∗
k′l if k ≥ k′,
0 if k < k′,
γd+1
k′k α
d∗
kl =
α
d∗
k′l if k ≥ k′,
0 if k < k′,
γdk′kα
d
kl =
α
d
k′l if k ≤ k′,
0 if k > k′,
γdk′kα
d∗
kl =
α
d∗
k′l if k ≤ k′,
0 if k > k′,
γd∗k′kα
d
kl = αd∗k′l, γd∗k′kα
d∗
kl = α
d∗∗
k′l
(always r = d+ 2p− 3).
3. Atoms in S2p−1
p . For n ≤ 2p− 1 the description of the category Snp is very simple. First, the
next fact is rather obvious.
Proposition 3.1. If n < 2p− 1, all indecomposable polyhedra in Snp are Moore spaces Md
k . In
particular, M2
k are atoms in S2p and there are no atoms in Snp if 2 < n < 2p− 1.
Proof is an easy induction. For n = 2 it is known. Suppose that 2 < n < 2p− 1 and the claim
is true for Sn−1p . We use Theorem 1.1 with m = 2n − 2. Then A consists of wedges of the sphere
S2n−2, while the spheres Sd (n ≤ d ≤ 2n − 2) and the Moore atoms Md
k (n < d ≤ 2n − 2)
form a set of additive generators of B. Note that in our case Mdr
k0 = 0 for n < d ≤ r ≤ 2n − 2,
exceptM2n−2,2n−2
00 . Therefore, the only new indecomposable polyhedra in Snp are the Moore spaces
M2n−1
k , which are not atoms.
Proposition 3.1 is proved.
Consider the category S
2p−1
p . Again we use Theorem 1.1 with m = 2n− 3 = 4p− 5. Now a set
of additive generators of A is
A =
{
S4p−4 = M4p−4
0 , S4p−5 = M4p−5
0 , M4p−5
k
}
,
and a set of additive generators of B is
B =
{
Sd = Md
0 (2p− 1 ≤ d ≤ 4p− 5), Md
k (2p− 1 < d ≤ 4p− 5)
}
.
The only nonzero values of Hosp(A,B), where A ∈ A, B ∈ B, are
M2p,4(p−1)
kl ' Z/p, with generators α(2p−1)∗
kl , k ∈ N, l ∈ N ∪ {0},
M2p−1,4(p−1)
0l ' Z/p with generators α2p−1
0l , l ∈ N ∪ {0},
M4p−5,4p−5
00 = Zp with generator γ4p−500 .
Therefore, the matrix F defining a morphism f : A → B (A ∈ A, B ∈ B) is a direct sum F ′ ⊕
⊕F ′′, where F ′′ is with coefficients fromM4p−5,4p−5
00 and F ′ is a block matrix (Fkl)k,l∈N∪{0}, where
Fkl is with coefficients fromM2p,4(p−1)
kl if k 6= 0 and F0l is with coefficients fromM2p−1,4(p−1)
0l . We
denote by Fk the horizontal stripe (Fkl)l∈N∪{0} with fixed k and by F l the vertical stripe (Fkl)k∈N∪{0}
with fixed l. Morphisms between objects from A and B act according to the rules (2.4). They imply
that two matrices F and G of such structure define isomorphic objects from E(M) if and only if
G′′ = TF ′′T ′ for some invertible matrices T, T ′ over Zp and F ′ can be transformed to G′ by a
sequence of the following transformations:
Fk 7→ TFk, where T is an invertible matrix over Z/p;
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 463
F l 7→ F lT ′, where T ′ is an invertible matrix over Z/p;
Fk 7→ Fk + UFk′ , where k′ > k or k′ = 0, k 6= 0 and U is any matrix of appropriate size over
Z/p;
F l 7→ F l + F l
′
U ′, where l′ < l and U ′ is any matrix of appropriate size over Z/p.
Using these transformations one can easily make the matrix F ′′ diagonal and reduce F ′ to a
matrix having at most one nonzero element in each row and in each column. Then the corresponding
object from E(M) splits into direct sum of objects given by (1 × 1)-matrices. The (1 × 1)-matrices
overM4p−5,4p−5
00 give Moore polyhedra M4p−4
t , which are not atoms (and belong to A). Therefore,
the atoms in S
2p−1
p are Ckl (k, l ∈ N ∪ {0}) corresponding to the (1 × 1)-matrices
(
α
(2p−1)∗
kl
)
if
k 6= 0 and to
(
α2p−1
0l
)
if k = 0. We call these polyhedra Chang atoms, in analogy with [2]. They are
defined by the cofibration sequences
M4p−4
l →M2p
k → Ckl →M4p−3
l →M2p+1
k if k 6= 0,
M4p−4
l → S2p−1 → C0l →M4p−3
l → S2p if k = 0.
(Ckl)
We can also present Chang atoms by their gluing diagrams, as in [2, 10, 11]:
C00 C0l Ck0 Ckl
4p− 3 • •
������������������������ •
������������������������ •
������������������������
pl
4p− 4 •
pl
•
2p •
pk
•
pk
2p− 1 • • • •
Here bullets correspond to cells, lines show the attaching maps and these maps are specified if
necessary.
Theorem 1.1 and cofibration sequences (Ckl) easily give the following values of the endomor-
phism rings of Chang atoms modulo the ideal J:
∆ = {(a, b) | a ≡ b (mod p) } ⊂ Zp × Zp for C00,
∆k = {(a, b) | a ≡ b (mod p) } ⊂ Zp × Z/pk for C0k and Ck0 (k 6= 0),
∆kl = {(a, b) | a ≡ b (mod p) } ⊂ Z/pk × Z/pl for Ckl (k 6= 0, l 6= 0).
Since all these rings are local and J2 = 0, the endomorphism rings of Chang atoms are local.
Therefore, these polyhedra are indeed indecomposable (hence atoms). Moreover, we can use the
unique decomposition theorem of Krull – Schmidt – Azumaya [1] (Theorem I.3.6) and obtain the final
result.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
464 YU. A. DROZD, P. O. KOLESNYK
Theorem 3.1. The atoms in S
2p−1
p are Chang atoms Ckl (k, l ∈ N ∪ {0}). Every polyhedron
from S
2p−1
p uniquely decomposes into a wedge of spheres, Moore polyhedra and Chang atoms.
In Section 5 we will need the whole endomorphism ring of the atom C = C00. Applying Hosp
to the sequence (C00) as below, we obtain the commutative diagram with exact columns and rows
ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 7
Theorem 3.1. The atoms in S
2p−1
p are Chang atoms Ckl (k, l ∈ N ∪ {0}). Every polyhedron
from S
2p−1
p uniquely decomposes into a wedge of spheres, Moore polyhedra and Chang atoms.
In Section 5 we will need the whole endomorphism ring of the atom C = C00. Applying Hosp
to the sequence (C00) as below, we obtain the commutative diagram with exact columns and rows
S2p �� S4p−3 �� C �� S2p−1 �� S4p−4
S4p−4
��
0 ��
��
0 ��
��
0 ��
��
0 ��
��
Zp
s ��
S2p−1
��
0 ��
��
0
��
�� pZp
��
��
Zp
s
��
1 ��
Z/p
��
C
��
0 ��
��
pZp
��
��
Hosp(C, C) ��
��
Zp
��
��
0
��
S4p−3
��
0 ��
��
Zp
1
��
s ��
Zp
��
��
0 ��
��
0
��
S2p Zp
s
�� Z/p �� 0 �� 0 �� 0
(3.1)
where s marks surjections. The central row and the central column, corresponding to the polyhedron
C, are easily calculated from all other values. It shows that Hosp(C, C) has no torsion, hence coin-
cides with ∆. Analogous calculations show that J(Ckl, Ckl) equals Z/p if k = 0 or l = 0 (but not
both) and (Z/p)2 if both k �= 0 and l �= 0.
Theorem 3.1 also gives a description of genera of p-primary polyhedra in S2p−1. Recall that a
genus is a class of polyhedra such that all their localizations are isomorphic (in the corresponding
localized categories). Certainly, if these polyhedra are p-primary, we only need to compare their p-
localizations. Equivalently, two polyhedra X, Y are in the same genus if and only if there is a wedge
of spheres W such that X ∨ W � Y ∨ W in S [12] (Theorem 2.5). Let g(X) be the number of
isomorphism classes of polyhedra in the genus of X. If Λ = Hos(X, X)/ tors(X), where tors(X)
is the torsion part of Hos(X, X), then Q Λ is a semi simple Q-algebra, so there is a maximal order
Γ ⊇ Λ in this algebra. Then Λ ⊇ mΓ for some positive integer m and g(X) = g(Λ) equals the
number of cosets
Im γ\(Γ/mΓ)×/(Λ/mΛ)×,
where R× denotes the group of invertible elements of a ring R and γ is the natural map Γ× →
→ (Γ/mΓ)× [12] (Section 3). If X = C0k or X = Ck0, then Λ = Z; if X = Ckt, then Λ = 0.
So g(X) = 1 for all these cases. For X = C this formula implies that g(C) = (p − 1)/2. If
ν ∈ Hosp(S
4p−4, S2p−1) is an element of order p, the polyhedra from the genus of C can be realized
as the cones C(c) of the maps S4p−4 cν−→ S2p−1 for 1 ≤ c ≤ (p − 1)/2.
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(3.1)
where s marks surjections. The central row and the central column, corresponding to the polyhedron
C, are easily calculated from all other values. It shows that Hosp(C,C) has no torsion, hence coin-
cides with ∆. Analogous calculations show that J(Ckl, Ckl) equals Z/p if k = 0 or l = 0 (but not
both) and (Z/p)2 if both k 6= 0 and l 6= 0.
Theorem 3.1 also gives a description of genera of p-primary polyhedra in S2p−1. Recall that a
genus is a class of polyhedra such that all their localizations are isomorphic (in the corresponding
localized categories). Certainly, if these polyhedra are p-primary, we only need to compare their p-
localizations. Equivalently, two polyhedra X, Y are in the same genus if and only if there is a wedge
of spheres W such that X ∨W ' Y ∨W in S [12] (Theorem 2.5). Let g(X) be the number of
isomorphism classes of polyhedra in the genus of X. If Λ = Hos(X,X)/ tors(X), where tors(X)
is the torsion part of Hos(X,X), then Q⊗Λ is a semisimple Q-algebra, so there is a maximal order
Γ ⊇ Λ in this algebra. Then Λ ⊇ mΓ for some positive integer m and g(X) = g(Λ) equals the
number of cosets
Im γ\(Γ/mΓ)×/(Λ/mΛ)×,
where R× denotes the group of invertible elements of a ring R and γ is the natural map Γ× →
→ (Γ/mΓ)× [12] (Section 3). If X = C0k or X = Ck0, then Λ = Z; if X = Ckt, then Λ = 0.
So g(X) = 1 for all these cases. For X = C this formula implies that g(C) = (p − 1)/2. If
ν ∈ Hosp(S
4p−4, S2p−1) is an element of order p, the polyhedra from the genus of C can be realized
as the cones C(c) of the maps S4p−4 cν−→ S2p−1 for 1 ≤ c ≤ (p− 1)/2.
4. Atoms in Sn
p for 2p ≤ n ≤ 4(p − 1). Let now 2p ≤ n ≤ 4(p − 1). We use Theorem 1.1
with m = n+ 2p− 3. Then A has a set of additive generators
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ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 465
A = {Sr (m ≤ r < 2n− 1), M r
l (m < r < 2n− 1, l ∈ N},
and B has a set of additive generators
B =
{
Sd (n ≤ d ≤ m), Md
k (n < d ≤ m, k ∈ N)
}
.
Morphisms ϕ : A → B, where A ∈ A, B ∈ B, are given by block matrices such that their blocks
have coefficients fromMdr
kl . Taking into consideration Definition 2.1, it is convenient to denote these
blocks as follows.
Definition 4.1. We introduce sets
E◦ =
{
edk (n < d ≤ 2(n− p) + 1, k ∈ N ∪ {0}), ed∗k (n ≤ d ≤ 2(n− p), k ∈ N), en0 , e
m
0
}
,
F◦ =
{
fdl (n < d ≤ 2(n− p) + 1, l ∈ N ∪ {0}), fd∗l (n ≤ d ≤ 2(n− p), l ∈ N), fn0
}
,
and consider a morphism ϕ : A → B, where A ∈ A, B ∈ B, as a block matrix (Φef )e∈E◦,f∈F◦ .
Namely,
the block Φedk,f
d
l
consists of coefficients at αdkl;
the block Φed∗k ,fdl
consists of coefficients at αd∗kl ;
the block Φedk,f
d∗
l
consists of coefficients at αd
∗
kl ;
the block Φed∗k ,fd∗l
consists of coefficients at αd
∗
∗
kl ;
the block Φem0 ,f
n
0
consists of coefficients at γm00.
Note that for n = 4(p− 1) we need not specially add em0 to E◦, since m = 2(n− p) + 1 in this
case.
We also denote by Φe for a fixed e ∈ E◦ the horizontal stripe (Φef )f∈F◦ and by Φf for a fixed
f ∈ F◦ the vertical stripe (Φef )e∈E◦ .
Note that the horizontal stripes Φedk
and Φ
e
(d+1)∗
k
have the same number of rows and the vertical
stripes Φfdl and Φf
(d+1)∗
l have the same number of columns. All blocks Φef defined above have
coefficients from Z/p, except Φem0 ,f
n
0
which has coefficients from Zp.
Using automorphisms of Sm we can make the block Φem0 ,f
n
0
diagonal with powers of p or zero
on diagonal. So we always suppose that it is of this shape and exclude this block from the matrix
Φ. Then we have to split the remaining part of the vertical stripe Φfn0 and, if n = 4(p − 1),
of the horizontal stripe Φem0
into several stripes, respectively, Φfn,s
0 and Φem,s
0
, where the indices
s ∈ N ∪ {∞} correspond to diagonal entries ps (setting p∞ = 0). Respectively, we modify the sets
E◦ and F◦. Namely, we denote
F = (F◦ \ {fn0 }) ∪ {fn,s0 | s ∈ N ∪ {∞}} ,
E = E◦ \ {em0 } if n < 4(p− 1), (4.1)
E = (E◦ \ {em0 }) ∪ {em,s0 | s ∈ N ∪ {∞}} if n = 4(p− 1).
Note that, if n = 4(p − 1), the number of rows in the horizontal stripe Φ
ed,s0
with s 6= ∞ equals
the number of columns in the vertical stripe Φfd,s0 . We split the sets E and F according to the upper
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
466 YU. A. DROZD, P. O. KOLESNYK
indices. Namely, Ed consists of all elements from E with the upper index d, d∗ or, if d = m, (m, s);
Fd consists of all elements from F with the upper index d, d∗ or, if d = n, (n, s). We define a linear
order on each Ed and Fd setting
edk < edk′ and ed∗k > ed∗k′ if k < k′, and edk < ed∗k′ for all k, k′;
if n = 4(p− 1), then em,s0 < em,s
′
0 < emk for s > s′ and any k ∈ N;
fdk < fdk′ and fd∗k > fd∗k′ if k < k′ or k > k′ = 0, and fdk < fd∗k′ for all k, k′;
fmk < fm,s0 < fm,s
′
0 for s < s′ and any k ∈ N.
The formulae (2.4) imply that two such block matrices Φ and Φ′ define isomorphic objects from
E(M) if and only if Φ can be transformed to Φ′ by a sequence of the following transformations:
Φe 7→ TeΦe, where Te are invertible matrices and Ted∗k = Ted+1
k
for all possible values of d, k;
Φf 7→ΦfT f , where T f are invertible matrices and T f
d∗
k = T f
d+1
k for all possible values of d, k;
if n = 4(p− 1), then, moreover, Tem,s
0
= T f
n,s
0 for all s ∈ N (not for s =∞);
Φe 7→ Uee′Φe′ if e′ < e, where Uee′ is an arbitrary matrix of the appropriate size;
Φf 7→ Φf ′Uf
′f if f ′ > f, where Uf
′f is an arbitrary matrix of the appropriate size.
These rules show that the classification of polyhedra in Snp actually coincides with the classi-
fication of representations of the bunch of chains X = {Ed,Fd, <,∼ | n ≤ d ≤ m} (cf. [6] or [7]
(Appendix B)), where the relation ∼ is defined by the exclusive rules:
ed∗k ∼ ed+1
k and fd∗k ∼ fd+1
k for n < d ≤ 2(n− p), k ∈ N,
and, if n = 4(p− 1),
em,s0 ∼ fn,s0 for s ∈ N (not for s =∞).
Thus the description of indecomposable representations given in [6, 7] implies a description of
indecomposable polyhedra from Snp . Recall the necessary combinatorics. We write e− f and f − e if
e ∈ Ed and f ∈ Fd (with the same d) and set |X| = E ∪ F.
Definition 4.2. (1) A word is a sequence w = x1r1x2r2 . . . xl−1rl−1xl, where xi ∈ |X|, ri ∈
∈ {−,∼} such that
a) ri 6= ri+1 for all 1 ≤ i < l − 1;
b) xirixi+1, 1 ≤ i < l, according to the definition of the relations ∼ and − given above;
c) if r1 = − (rl−1 = −), then x1 � y for all y ∈ |X| (respectively, xl � y for all y ∈ |X|).
We say that l is the length of the word w and write l = lnw.
(2) For a word w as above we denote by E(w) = { i | 1 ≤ i ≤ l, xi ∈ E } and F(w) = {i | 1 ≤
≤ i ≤ l, xi ∈ F}.
(3) The inverse word w∗ of the word w is the word xlrl−1xl−1 . . . r2x2r1x1.
(4) A word w is said to be a cycle if r1 = rl−1 =∼ and xl − x1. Then we set rl = −, xi+ql = xi
and ri+ql = ri for all q ∈ Z (in particular, r0 = −).
(5) The k th shift of a cycle w, where k is an even integer, is the cycle w[k] = xk+1rk+1 . . . rk−1xk
(obviously, it is enough to consider 0 ≤ k < l).
(6) A cycle w is said to be non periodic if w 6= w[k] for 0 < k < l.
(7) For a cycle w and an integer 0 < k < l we denote by ν(k,w) the number of even integers
0 < i < k such that both xi and xi−1 belong either to E or to F.
Note that, since x � x for all x ∈ |X|, there are no symmetric words and symmetric cycles in the
sense of [7] (Appendix B).
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ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 467
To words and cycles correspond indecomposable representations of the bunch of chains X called
strings and bands. We describe the corresponding matrices Φ (recall that we have already excluded
the part Φemfn).
Definition 4.3. (1) If w is a word, the corresponding string matrix Φ(w) is constructed as
follows:
its rows are labelled by the set E(w) and its columns are labelled by the set F(w);
the only nonzero entries are those at the places (i, i+ 1) if ri = − and i ∈ E(w) and (i+ 1, i)
if ri = − and xi ∈ F(w); they equal 1.
We denote the corresponding polyhedron by A(w) and call it a string polyhedron whenever it does
not coincide with a sphere, a Moore or a Chang polyhedron1.
(2) If w is a non periodic cycle, z ∈ N and π 6= t is a unital irreducible polynomial of degree
v from (Z/p)[t], the band matrix Φ(w, z, π) is a block matrix, where all blocks are of size zv × zv,
constructed as follows:
its horizontal stripes are labelled by the set E(w) and its vertical stripes are labelled by the set
F(w);
the only nonzero blocks are those at the places (i, i+ 1) if ri = − and i ∈ E(w) and (i+ 1, i) if
ri = − and i ∈ F(w) (note that here i = l is also possible);
these nonzero blocks equal Izv (the identity zv × zv matrix), except the block at the place (l1)(
if l ∈ E(w)
)
or (1l)
(
if l ∈ F(w)
)
which is the Frobenius matrix with the characteristic polynomial
πv. If π = t − c is linear, we replace the Frobenius matrix by the Jordan z × z block with the
eigenvalue c.
We denote the corresponding polyhedron by A(w, z, π) and call it a band polyhedron2.
Using these notions, we obtain the following description of polyhedra in the category Snp .
Theorem 4.1. (1) All string and band polyhedra are indecomposable and every indecomposable
polyhedron from Snp , except spheres, Moore and Chang polyhedra, is isomorphic to a string or band
polyhedron.
(2) The only isomorphisms between string and band polyhedra are the following:
A(w) ' A(w∗);
A(w, z, π) ' A(w∗, z, π);
A(w, z, π) ' A(w[k], z, π∗), where π∗ = π if ν(k,w) is even and π∗(t) = tzπ(0)−1π(1/t) if
ν(k,w) is odd3.
(3) Endomorphism rings of string and band polyhedra are local, hence every polyhedron from
Snp uniquely decomposes into a wedge of spheres, Moore and Chang polyhedra, and string and band
polyhedra.
(4) A string or band polyhedron is an atom in Snp if and only if the corresponding word contains
at least one letter from Ed and at least one letter from F2(n−p)+1.
Note that in this case we can simplify the writing of the words, since for every x ∈ |X| there
is at most one element y ∈ |X| such that x ∼ y and then x − y is impossible. Hence we can
omit all symbols − and write x instead of x ∼ y. For instance, edkf
d−1
l e
(d−2)∗
k′ fd−1l′ means edk ∼
∼ e
(d−1)∗
k − fd−1l ∼ f
(d−2)∗
l − ed−2k′ ∼ e
(d−1)∗
k′ − fd−1l′ ∼ f
(d−2)∗
l′ . One can prove that there can be
1 The words consisting of one letter x correspond to spheres, the words of the form x ∼ y correspond to Moore
polyhedra, the words that only have one symbol ‘−’ correspond to Chang polyhedra, and these are all exceptions.
2 Band polyhedra never coincide with spheres, Moore or Chang polyhedra.
3 If π = tv + a1t
v−1 + . . .+ av−1t+ av, then π∗ = tv + a−1
v (av−1t
v−1 + . . .+ a1t+ 1).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
468 YU. A. DROZD, P. O. KOLESNYK
at most one place in a word w where a fragment em,s ∼ fn,s or fn,s0 ∼ em,s0 occurs; moreover, if it
occurs, w cannot be a cycle.
Example 4.1. We give several examples of string and band polyhedra and their gluing diagrams.
In these examples we suppose that p = 3.
(1) The “smallest” possible string atoms are for n = 6. They have 3 cells and are given by
the words e6∗k f
7
0 or e60f
6∗
l . The smallest band atoms have 4 cells. They are A(w0, 1, t ∓ 1), where
w0 = e7kf
7
l . Here are their gluing diagrams:
11 • •
3l
•
3l
10 • •
9
8
7 •
3k
������������������������������ •
3k
������������������������������
6 • •
������������������������������ •
±1
������������������������������
(2) More complicated band atoms are A(w0, 1, t
2+1) and A(w0, 2, t∓1). Their gluing diagrams
are
11 •
3l
•
3l
•
3l
•
3l
10 • • • •
9
8
7 •
3k
•
3k
•
3k
•
3k
6 •
−1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~
•
����������������������������� •
±1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ •
±1
The nontrivial attachments of cells of dimension 10 come, respectively, from the Frobenius matrix(
0 −1
1 0
)
and the Jordan block
(
±1 1
0 ±1
)
.
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ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 469
(3) For the maximal value n = 8 the smallest atoms contain 4 cells. They are given by the words
e80f
8,s
0 f110 and have the gluing diagrams
15 •
14
13
12 •
3s
11 •
������������������������������
10
9
8 •
������������������������������
(4) The band atoms for n = 8 are rather complicated and cannot be “small”. For instance, one of
the smallest is A(w, 1, t∓ 1), where w = e8∗k1f
9∗
l1
e10∗k2 f
11
l2
e10k3f
9
l3
. The gluing diagram for this atom is
15 •
14 •
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, •
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
13 • •
12 •
11 •
������������������������������
10 • •
9 •
������������������������������ •
������������������������������
8 •
±1ooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooo
(the powers of 3 near vertical lines are omitted).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
470 YU. A. DROZD, P. O. KOLESNYK
(5) Finally, we give an example of an atom having exactly one cell of each dimension (we do
not precise the corresponding word, since it can be easily restored).
15 •
+++++++++++++++++++++++++++++++
14 •
13 •
+++++++++++++++++++++++++++++++
12 •
11 •
10 •
�������������������������������
9 •
8 •
�������������������������������
Another atom with this property is the properly shifted S-dual of this one in the sense of [15]
(Chapter 14).
One can also calculate genera of p-primary polyhedra for 2p ≤ n ≤ 4(p− 1). Namely, let Λ(X)
denotes the ring Hos(X,X)/ tors(X). We call the end x1 or xl of a word w spherical if it of the
form ed0 or fd0 . Note that these letters can only occur at an end of the word since they are not related
by ∼ to any letter. It is rather easy to verify that Λ(X) = 0 if X is a band polyhedron, while for a
string polyhedron X = A(w)
Λ(X) =
0 if w has no spherical ends,
Z if one end of w is spherical,
∆ if both ends of w are spherical.
Hence, we obtain the following result.
Corollary 4.1. If X is a band or string polyhedron, then g(X) = 1, except the case when
X = A(w) and both ends of the word w are spherical. In the latter case g(X) = (p− 1)/2.
5. Case n > 4(p− 1). For n = 4p− 3 we set m = 6p− 5 = n+ 2p− 2 and q = 2(n− 1) =
= n + 4p − 5 = m + 2p − 3. Then A contains Moore polyhedra M q
k (including Sq = M q
0 ) and
B contains the shifted Chang polyhedron Cm = C00[2p − 2]. Let Nk = Hosp(M
q
k , C
m). Applying
Hosp(M
q
k , ) to the cofibre sequence
0→ Sm−1 → Sn → Cm → Sm → Sn+1
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ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY 471
we get an exact sequence
0→ Z/p λ−→ Nk
µ−→ Z/p→ 0.
Thus #(Nk) = p2. On the other hand, applying Hosp( , C) to the cofibre sequence (Edk) of Section 2,
we get an exact sequence
N0
pk−→ N0
η−→ Nk → 0.
Therefore the map η is an isomorphism. Setting k = 1, we see that pN0 = 0, hence N0 ' Z/p×Z/p
and Nk ' Z/p × Z/p for all k. We denote by λk a generator of Nk which is in Imλ and by µk a
generator of Nk such that µ(µk) 6= 0.
Analogous observations show that the generator of the cyclic group Mqq
kl = Hosp(M
q
l ,M
q
k )
induces an isomorphism Nk → Nl if k ≥ l > 0 and zero map if 0 < k < l. On the other
hand, the diagram (3.1) implies that an element (a, b) of the ring ∆ = Hosp(C,C) acts on Nk as
multiplication by a
(
recall that a ≡ b (mod p)
)
. Therefore, a map ϕ : A → B, where A is a wedge
of Moore polyhedra M q
k and B is a wedge of Chang polyhedra Cm can be considered as a block
matrix Φ = (Φik)k∈N∪{0}
i=1,2
, where all blocks are with coefficients from Z/p and both horizontal stripes
Φ1, Φ2 have the same number of rows. Namely, Φ1k consists of coefficients at λk and Φ2k consists
of coefficients at µk. Two such matrices define isomorphic objects from E(M) if and only if one of
them can be transformed to the other by a sequence of the following transformations:
Φ1 7→ TΦ1 and Φ2 7→ TΦ2 with the same invertible matrix T ;
Φk 7→ ΦkT k for some invertible matrix T k;
Φk 7→ Φk + ΦlUlk for any matrix Ulk of the appropriate size, where l > k or l = 0 < k.
It is well-known that this matrix problem is wild, i.e., contains the problem of classification of pairs
of linear maps in a vector space; hence, a problem of classification of representations of any finitely
generated algebra over the field Z/p (cf. [9], Section 5). Namely, consider the case when the matrix
Φ = Φ(F,G) is of the form
I 0 0
0 I 0
F I 0
G 0 I
.
Here I is a unit matrix of some size, F and G are arbitrary square matrices of the same size; line
show the subdivision of Φ into blocks Φik (there are only two vertical stripes). One easily checks
that Φ(F,G) and Φ(F ′, G′) define isomorphic objects if and only if there is an invertible matrix T
such that F ′ = TFT−1 and G′ = TGT−1. So we obtain the following result.
Theorem 5.1. The classification of p-local polyhedra in Snp for n > 4(p− 1) is a wild problem.
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Received 21.05.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
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| resource_txt_mv | umjimathkievua/ef/ddbe515270c3925ba7370b1ff17219ef.pdf |
| spelling | umjimathkievua-article-21482019-12-05T10:25:15Z Atoms in the p-localization of Stable Homotopy Category Атоми в $p$-локалізації стабльної гомотопічної категорії Drozd, Yu. A. Kolesnyk, P. O. Дрозд, Ю. А. Колесник, П. О. We study p-localizations, where p is an odd prime, of the full subcategories \( {\mathcal{S}}^n \) of stable homotopy category formed by CW-complexes with cells in n successive dimensions. Using the technique of triangulated categories and matrix problems, we classify the atoms (indecomposable objects) in \( {\mathcal{S}}_p^n \) for n ≤ 4(p − 1) and show that, for n > 4(p − 1), this classification is wild in a sense of the representation theory. Вивчаються $p$-локалiзацiї (де $p$ — непарне просте число) повних підкатегорій ${\mathcal{S}}^n$ стабільної гомотопічної категорії, що складається з CW-комплексів із клітинами в $n$ послідовних розмірностях. Застосовуючи техніку триангульованих категорій та матричні задачі, ми наводимо класифікацію атомів (нерозкладних об'єктів) у ${\mathcal{S}}_p^n$ Для $n ≤ 4(p − 1)$ i показуємо, що для $n > 4(p — 1)$ ця класифікація є дикою у сенсі теорії зображень. Institute of Mathematics, NAS of Ukraine 2014-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2148 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 4 (2014); 458–472 Український математичний журнал; Том 66 № 4 (2014); 458–472 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2148/1298 https://umj.imath.kiev.ua/index.php/umj/article/view/2148/1299 Copyright (c) 2014 Drozd Yu. A.; Kolesnyk P. O. |
| spellingShingle | Drozd, Yu. A. Kolesnyk, P. O. Дрозд, Ю. А. Колесник, П. О. Atoms in the p-localization of Stable Homotopy Category |
| title | Atoms in the p-localization of Stable Homotopy Category |
| title_alt | Атоми в $p$-локалізації стабльної гомотопічної категорії |
| title_full | Atoms in the p-localization of Stable Homotopy Category |
| title_fullStr | Atoms in the p-localization of Stable Homotopy Category |
| title_full_unstemmed | Atoms in the p-localization of Stable Homotopy Category |
| title_short | Atoms in the p-localization of Stable Homotopy Category |
| title_sort | atoms in the p-localization of stable homotopy category |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2148 |
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