DOI 10.1007/s11253-014-0949-0 Ukrainian Mathematical Journal, Vol. 66, No. 4, September, 2014 (Ukrainian Original Vol. 66, No. 4, April, 2014)

ATOMS IN THE p-LOCALIZATION OF STABLE HOMOTOPY CATEGORY Yu. A. Drozd and P. O. Kolesnyk

UDC 512.5

We study p -localizations, where p is an odd prime, of the full subcategories Sn 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 Sn p for n  4(p − 1) and show that, for n > 4(p − 1), this classification is wild in a sense of the representation theory.

Introduction The problem of classification of the homotopy types of polyhedra (finite CW-complexes) has a long history. It is well known that this problem becomes much simpler if we consider the stable situation, i.e., identify two polyhedra with homotopy equivalent (iterated) suspensions. This leads to the notion of stable homotopy category and stable homotopy equivalence. A classification of this sort was realized for polyhedra of low dimensions by several authors. A good survey of these results can be found in the paper by Baues [2]. Unfortunately, this 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 containing the 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 to be 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 a sense of [9]). In particular, it gives simplified proofs of the results from [3–5]. In [11], this technique gave new results on the classification of polyhedra with torsion-free homologies. The main difficulties in the stable homotopy classification are related to the 2-components of homotopy groups. This 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 the present paper we use the technique of [10, 11] to classify p-local polyhedra that have cells only in n successive dimensions for n  4(p − 1). Similar results were obtained by Henn [13] who used a different approach. Our description seems to be more straightforward and more visual. It gives an 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 and, hence, the obtained results are, in a certain 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 a sense that there is an upper bound for the number of cells in indecomposable polyhedra (atoms); actually, atoms have at most four cells. Section 4 is the main part of the paper. Here, we describe polyhedra for 2p  n  4(p − 1). The result is presented in terms of strings and bands, which is usual in the modern representation theory. String and band polyhedra are defined by some combinatorial invariant (a word) and, in the band case, an irreducible polynomial over the residue field Z/p. In the representation theory this description is called tame. Finally, in Section 5, we prove that the classification becomes wild for n > 4(p − 1). Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 4, pp. 458–472, April, 2014. Original article submitted May 21, 2013. 514

0041-5995/14/6604–0514

c 2014 �

Springer Science+Business Media New York

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The description obtained by matrix methods is local, just as the description of [13]. By using the results of [12] we also obtain a global description of p-primary polyhedra. Fortunately, it almost coincides with the local description, except rare special cases when one local object gives rise to (p − 1)/2 global objects.

The first author expresses his gratitude 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. This is an additive category and the morphism groups in this category are � � Hos(X, Y ) = lim Hot X[k], Y [k] , −! 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 ⇡nS (X) = Hos(S n , X) are torsion groups for n > dim X. It is convenient to formally add to S the “negative shifts” X[−k], k 2 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 2 Z. Then S becomes a triangulated category, where the suspension plays the role of shift and the exact triangles are cofiber sequences X ! Y ! Z ! X[1] (in S, they are the same as fiber sequences). In what follows, 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 2 Z, of polyhedra having cells only in at most n successive dimensions or, which is 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 not greater than 2n − 1. We denote the full subcategory n of Sn consisting of these polyhedra by S . Moreover, if two polyhedra of this kind are isomorphic in S, then they n are homotopy equivalent. Following Baues [2], an object from Sn is called an atom if it belongs to S , does not belong to Sn−1 , and is indecomposable (into a wedge of noncontractible polyhedra). Recall that the p-localization of an additive category C is a category Cp such that Ob Cp = Ob C and HomCp (A, B) = Zp ⌦ HomC (A, B), where Zp ⇢ Q is a subring

o na� � � a, b 2 Z, p - b . 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 a sense of [14]. Every space of this kind can be regarded as an image in Sp of a p-primary polyhedron, i.e., a polyhedron X such that the map pk 1X for some k can be factored through a wedge of spheres [8]. To study the categories Snp , we use the technique of bimodule categories, as in [11]. We recall the corresponding notions.

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Definition 1.1 (cf. [9], Section 4). Let A and B be additive categories and let 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: S 1. Ob E(M) = A2Ob A M(A, B). B2Ob B

2. If u 2 M(A, B), v 2 M(A0 , B 0 ), then

� HomE(M) (u, v) = (f, g) | f : A ! A0 , g : B ! B 0 , gu = vf

(both these elements are from M(A, B 0 )). E(M) is also an additive category. Note that we consider only bipartite bimodules in a sense of [9]. Usually, we choose a set of additive generators of A and B, i.e., sets {A1 , A2 , . . . , As } ⇢ Ob A and {B , B , . . . , Br } ⇢ Ob B such that every object from A (respectively, from B) is isomorphic to the direct sum Lr L1s 2 j=1 kj Aj (respectively, i=1 li Bi ). Then an object of E(M) can be represented as a block matrix F = (Fij ), where Fij is a matrix of size li ⇥ kj with coefficients from M(Aj , Bi ). If we represent morphisms in the analogous matrix form, then the action of morphisms on the elements from M is represented by the ordinary 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 formed by (m − 1)-connected polyhedra of dimension at most 2n − 2 (respectively, by (n − 1)-connected polyhedra of dimension at most m). Consider an 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 (↵, β) : f ! f 0 such that ↵ factors through f and β factors n through f 0 . Also let J be the ideal of Sp formed by 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 n n n E(M)/I ' Sp /J. Moreover, J2 = 0. Hence, the isomorphism classes of the categories Sp and Sp /J are identical. We also note 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(S l , S k ) are Hosp (S k+qs , S k ) = Z/p, where 1  s < p and qs = 2s(p − 1) − 1 [16]. 2. Moore Polyhedra pk

The only atoms in S2p are Moore atoms Mk (k 2 N) which are cones of the maps S 2 −! S 2 . We denote their d-dimensional suspensions Mk [d − 3] by Mkd and call them Moore polyhedra. For the sake of unification, r d we denote S d by M0d . We need to know the morphism groups Mdr kl = Hosp (Ml , Mk ). It is always supposed that d,d+2p−3 = Z/p, and Mdr / {d, d + 2p − 3}. d − 1  r < d + 2p − 1. Clearly, Mdd 00 = Zp , M00 00 = 0 for r 2 If k > 0, then, for the cofiber sequences pk

pk

S d−1 −! S d−1 ! Mkd ! S d −! S d , dr we easily conclude that Mdr 0k = Mk0 = 0, except the cases k Md,d−1 ' Mdd 0k ' Z/p , k0

Md,d+2p−3 ' Md,d+2p−3 ' Md,d+2p−4 ' Md,d+2p−2 ' Z/p. k0 0k k0 0k

(Edk )

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r The values of Mdr kl for k, l 2 N, d − 1  r < d + 2p − 1, can be obtained if we apply Hosp (Ml , ) to the d cofiber sequences (Ek ). This gives the exact sequences pk

pk

dr dr −! Md−1,r ! Mdr Md−1,r kl ! M0l −! M0l , 0l 0l

whence we get

Mdr kl =

8 Z/pmin(k,l) > > > > > > >
> > Z/p ⊕ Z/p > > > > > : 0,

for r 2 {d − 1, d} , for r 2 {d + 2p − 2, d + 2p − 4} , for r = d + 2p − 3,

(2.1)

in the other cases.

The only nontrivial value in this case is obtained for r = d + 2p − 3: It is necessary to know that the exact sequence ↵

β

0 ! Z/p − ! Md,d+2p−3 − ! Z/p ! 0 kl

(2.2)

splits. This sequence indeed splits for k = 1 because the middle term is a module over Mdd 11 = Z/p. For k > 1, splits. The commutative diagram we suppose that the sequence for Md,d+2p−3 k−1,l pk

S d−1 −−−−! S d−1 −−−−! Mkd −−−−! ? ? ? ? ? ? py 1y y pk−1

pk

S d −−−−! ? ? py pk−1

Sd ? ? 1y

(2.3)

d −−−−! S d −−−−! S d S d−1 −−−−! S d−1 −−−−! Mk−1

induces the commutative diagram −−−−! Z/p −−−−! 0 0 −−−−! Z/p −−−−! Md,d+2p−3 kl ? ? ? ? ? ? 1y 0 y y

0 −−−−! Z/p −−−−! Md,d+2p−3 −−−−! Z/p −−−−! 0. k−1,l

Since the second row splits, the first row also splits. Therefore, sequence (2.2) splits for all values of k and l. Definition 2.1. We fix generators of the groups Mdr kl and denote, for r = d + 2p − 3, d⇤

d+1,r+1 by ↵kl⇤ (k, l 2 N), the generator of Mkl lying in the image of the map ↵ from (2.2); d (k, l 2 N [ {0}), the generator of Mdr which is not in Im ↵; by ↵kl kl ⇤

d (k 2 N [ {0}, l 2 N), the generator of Md,r+1 ; by ↵kl kl d+1,r d⇤ (k 2 N, l 2 N [ {0}), the generator of Mkl ; by ↵kl d (k, l 2 N [ {0}), the generator of Mdd ; by γkl kl d⇤ (k 2 N, l 2 N [ {0}), the generator of Md+1,d . by γkl kl

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Note that all these morphisms are actually induced by the maps S r ! S d . By using diagrams of the form (2.3), one can easily show that these generators can be chosen so that 8 d⇤ 8 ⇤ d <↵kl⇤0 if l  l0 , <↵kl if l  l0 , 0 ⇤ ⇤ d⇤ r+1 d r+1 ↵kl γll0 = ↵kl γll0 = : : 0 if l > l0 , 0 if l > l0 , d r γll0 ↵kl

=

8 d <↵kl 0 :

0

d⇤⇤ γkd+1 0 k ↵kl

d γkd0 k ↵kl

if l ≥ l0 or l = 0,

if 0 < l < l0 , ⇤

d r⇤ d ↵kl γlk0 = ↵kk 0,

=

=

8 d⇤ <↵k⇤0 l :

0

8 <↵kd0 l :

0

if k ≥ k 0 , if k < k 0 , if k  k 0 , if k > k 0 ,

d γkd⇤0 k ↵kl = ↵kd⇤0 l ,

d⇤ r ↵kl γll0

=

d⇤

8 d⇤ <↵kl 0 :

0

if l ≥ l0 or l = 0,

if 0 < l < l0 ,

d⇤ r⇤ ↵kl⇤ γlk 0 = ↵ kk0 ,

d⇤ γkd+1 0 k ↵kl

d⇤ γkd0 k ↵kl

⇤

=

=

8 <↵kd⇤0 l :

0

8 ⇤ <↵kd0 l :

0

(2.4) if k ≥ k 0 , if k < k 0 , if k  k 0 , if k > k 0 ,

d⇤

d γkd⇤0 k ↵kl = ↵k⇤0 l

(here, we always have 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 following fact is quite obvious: Proposition 3.1. If n < 2p − 1, then all indecomposable polyhedra in Snp are Moore spaces Mkd . In particular, Mk2 are atoms in S2p and there are no atoms in Snp if 2 < n < 2p − 1.

Proof. The proposition is proved by an evident induction. For n = 2, this fact is known. Suppose that 2 < n < 2p − 1 and the claim is true for Sn−1 . We use Theorem 1.1 with m = 2n − 2. Then A consists of p 2n−2 d , while the spheres S (n  d  2n − 2) and the Moore atoms Mkd (n < d  2n − 2) wedges of the sphere S form a set of additive generators of B. Note that, in our case Mdr k0 = 0 for n < d  r  2n − 2, except 2n−2,2n−2 . Therefore, the only new indecomposable polyhedra in Snp are the Moore spaces Mk2n−1 but they are M00 not atoms. Proposition 3.1 is proved. Consider the category S2p−1 . We again use Theorem 1.1 with m = 2n − 3 = 4p − 5. Hence, a set of additive p generators of A is o n A = S 4p−4 = M04p−4 , S 4p−5 = M04p−5 , Mk4p−5 , and a set of additive generators of B is

n o B = S d = M0d (2p − 1  d  4p − 5), Mkd (2p − 1 < d  4p − 5) .

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The only nonzero values of Hosp (A, B), where A 2 A and B 2 B, are 2p,4(p−1)

Mkl

(2p−1)⇤

' Z/p with generators ↵kl

2p−1,4(p−1)

M0l

, k 2 N, l 2 N [ {0},

2p−1 ' Z/p with generators ↵0l , l 2 N [ {0},

4p−5 M4p−5,4p−5 = Zp with a generator γ00 . 00

Therefore, the matrix F specifying a morphism f : A ! B (A 2 A, B 2 B) is the direct sum F 0 ⊕ F 00 , and F 0 is a block matrix (Fkl )k,l2N[{0} , where Fkl is with where F 00 is with coefficients from M4p−5,4p−5 00 2p,4(p−1)

2p−1,4(p−1)

if k 6= 0 and F0l is with coefficients from M0l . By Fk , we denote coefficients from Mkl l the horizontal stripe (Fkl )l2N[{0} with fixed k. By F , we denote the vertical stripe (Fkl )k2N[{0} with fixed l. Morphisms between the objects from A and B act according to rules (2.4). They imply that two matrices F and G of this structure define isomorphic objects from E(M) if and only if G00 = T F 00 T 0 for some invertible matrices T, T 0 over Zp and F 0 can be transformed into G0 by a sequence of the following transformations: Fk 7! T Fk , where T is an invertible matrix over Z/p; F l 7! F l T 0 , where T 0 is an invertible matrix over Z/p; Fk 7! Fk + U Fk0 , where k 0 > k or k 0 = 0, k 6= 0 and U is any matrix of appropriate size over Z/p; 0

F l 7! F l + F l U 0 , where l0 < l and U 0 is any matrix of appropriate size over Z/p. By using these transformations, one can easily diagonalize the matrix F 00 and reduce F 0 to a matrix with at most one nonzero element in each row and in each column. Then the corresponding object from E(M) splits give Moore into a direct sum of objects given by (1 ⇥ 1)-matrices. The (1 ⇥ 1)-matrices over M4p−5,4p−5 00 are C (k, l 2 N [ {0}) polyhedra Mt4p−4 that are not atoms (and belong to A). Therefore, the atoms in S2p−1 kl � 2p−1 � p � (2p−1)⇤ � for k 6= 0 and to ↵0l for k = 0. These polyhedra are corresponding to the (1 ⇥ 1)-matrices ↵kl called Chang atoms by analogy with [2]. They are defined by the cofibration sequences Ml4p−4 ! Mk2p ! Ckl ! Ml4p−3 ! Mk2p+1 Ml4p−4

!S

2p−1

! C0l !

Ml4p−3

!S

2p

for k 6= 0,

(Ckl )

for k = 0.

We can also represent Chang atoms by their gluing diagrams, as in [2, 10, 11]:

4p − 3

C00

C0l

•

•

pl

Ckl

•

•

•

4p − 4

pk

•

•

pl

•

2p 2p − 1

Ck0

• •

pk

• •

Here, the bullets correspond to the cells, the lines show the attaching maps, and (if necessary) these maps are specified.

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Theorem 1.1 and cofibration sequences (Ckl ) readily give the following values of the endomorphism rings of Chang atoms modulo the ideal J: ∆ = { (a, b) | a ⌘ b (mod p) } ⇢ Zp ⇥ Zp ∆k = { (a, b) | a ⌘ b (mod p) } ⇢ Zp ⇥ Z/pk

for C0k

∆kl = { (a, b) | a ⌘ b (mod p) } ⇢ Z/pk ⇥ Z/pl

for C00 , and

for Ckl

(k 6= 0),

Ck0

(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 (and, hence, they are atoms). Moreover, we can use the Krull–Schmidt– Azumaya unique decomposition theorem [1] (Theorem I.3.6) to obtain the final result. are Chang atoms Ckl (k, l 2 N [ {0}). Every polyhedron from S2p−1 is Theorem 3.1. The atoms in S2p−1 p p uniquely decomposed into a wedge of spheres, Moore polyhedra, and Chang atoms. In Section 5, we need the entire endomorphism ring of the atom C = C00 . Applying Hosp to the sequence (C00 ) as below, we arrive at the commutative diagram (3.1) with exact columns and rows, where s marks surjections. The central row and central column corresponding to the polyhedron C are easily calculated from all other values. This shows that Hosp (C, C) has no torsion and, hence, coincides with ∆. Analogous calculations demonstrate that J(Ckl , Ckl ) is equal to Z/p for k = 0 or l = 0 (but not both) and (Z/p)2 for both k 6= 0 and l 6= 0. S 2p

S 4p−4

0

✏ ✏

S 2p−1

0

✏ ✏

C

0

✏ ✏

S 4p−3

0

✏ ✏

S 2p

Zp

S 4p−3 o

C o

0 /

0 /

0

1

✏ ✏

pZp /

✏

s s

/

Hosp (C, C)

✏

Z/p

0 /

✏

0

✏

0 /

✏ /

(3.1) ✏

Zp /

Zp /

Z/p /

✏

✏

1

Zp /

/

✏

s

Zp /

Zp / s

✏

pZp /

S 4p−4 o

0 /

✏ ✏

/

S 2p−1 o

0

/

✏

/

0 /

✏

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, then we only need to compare their p-localizations. Equivalently,

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two polyhedra X, Y are of 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 ⇤ = Hos(X, X)/ tors(X), where tors(X) is the torsion part of Hos(X, X), then Q ⌦ ⇤ is a semisimple Q-algebra and, therefore, there is a maximal order Γ ◆ ⇤ in this algebra. Then ⇤ ◆ mΓ for some positive integer m and g(X) = g(⇤) is equal to the number of cosets Im γ\(Γ/mΓ)⇥ /(⇤/m⇤)⇥ , 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 ⇤ = Z. If X = Ckt , then ⇤ = 0. Thus, g(X) = 1 for all these cases. For X = C this formula implies that g(C) = (p − 1)/2. If ⌫ 2 Hosp (S 4p−4 , S 2p−1 ) is an element of c⌫ order p, then the polyhedra from the genus of C can be realized as the cones C(c) of the maps S 4p−4 −! S 2p−1 for 1  c  (p − 1)/2. 4. Atoms in Sn p for 2p  n  4(p − 1) Now let 2p  n  4(p − 1). We use Theorem 1.1 with m = n + 2p − 3. Then A has a set of additive generators A = {S r (m  r < 2n − 1), Mlr (m < r < 2n − 1, l 2 N}, and B has a set of additive generators o n B = S d (n  d  m), Mkd (n < d  m, k 2 N) . The morphisms ' : A ! B, where A 2 A, B 2 B, are given by block matrices such that their blocks have coefficients from Mdr kl . In view of Definition 2.1, it is convenient to denote these blocks as follows: Definition 4.1. We introduce sets n o n m E◦ = edk (n < d  2(n − p) + 1, k 2 N [ {0}), ed⇤ (n  d  2(n − p), k 2 N), e , e 0 0 , k n o F◦ = fld (n < d  2(n − p) + 1, l 2 N [ {0}), fld⇤ (n  d  2(n − p), l 2 N), f0n ,

and consider a morphism ' : A ! B, where A 2 A and B 2 B, as a block matrix (Φef )e2E◦ ,f 2F◦ . Namely, d ; the block Φed ,f d consists of the coefficients of ↵kl k

l

d⇤ ; the block Φed⇤ ,f d consists of the coefficients of ↵kl k

l

⇤

d ; the block Φed ,f d⇤ consists of the coefficients of ↵kl k

l

d⇤

the block Φed⇤ ,f d⇤ consists of the coefficients of ↵kl⇤ ; k

l

m n consists of the coefficients of γ the block Φem 00 . 0 ,f0 ◦ Note that, for n = 4(p − 1), it is not necessary to specially add em 0 to E because m = 2(n − p) + 1 in this case.

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We also denote by Φe , for a fixed e 2 E◦ , the horizontal stripe (Φef )f 2F◦ and by Φf , for a fixed f 2 F◦ , the vertical stripe (Φef )e2E◦ . d Note that the horizontal stripes Φed and Φe(d+1)⇤ have the same number of rows and the vertical stripes Φfl k

k

(d+1)⇤

and Φ have the same number of columns. All blocks Φef defined above have coefficients from Z/p, except n with coefficients from Zp . the block Φem 0 ,f0 n diagonal with powers of p or zeros in By using the automorphisms of S m , one can make the block Φem 0 ,f0 the diagonal. Thus, we always suppose that it is of the indicated shape and exclude this block from the matrix Φ. n Further, it is necessary to split the remaining part of the vertical stripe Φf0 [and if n = 4(p − 1), then of the n,s ] into several stripes Φf0 and Φem,s , respectively, where the indices s 2 N [ {1} correhorizontal stripe Φem 0 0 spond to the diagonal entries ps (by setting p1 = 0). In the same way, we modify the sets E◦ and F◦ . Namely, we denote fl

F = (F◦ \ {f0n }) [ { f0n,s | s 2 N [ {1} } , E = E◦ \ {em 0 }

if

(4.1)

n < 4(p − 1),

m,s E = (E◦ \ {em 0 }) [ { e0 | s 2 N [ {1} }

if

n = 4(p − 1).

Note that if n = 4(p − 1), then the number of rows in the horizontal stripe Φed,s with s 6= 1 is equal to the 0

d,s

number of columns in the vertical stripe Φf0 . We split the sets E and F according to the superscripts. Namely, Ed consists of all elements from E with the superscript d, d⇤ [if d = m, then (m, s)] and Fd consists of all elements from F with the superscript d, d⇤ [if d = n, then (n, s)]. We define a linear ordering on each Ed and Fd by setting d⇤ 0 d d⇤ 0 edk < edk0 and ed⇤ k > ek0 if k < k and ek < ek0 for all k, k ; 0

0 < em,s < em if n = 4(p − 1), then em,s 0 0 k for s > s and any k 2 N;

fkd < fkd0 and fkd⇤ > fkd⇤0 if k < k 0 or k > k 0 = 0 and fkd < fkd⇤0 for all k, k 0 ; 0

fkm < f0m,s < f0m,s for s < s0 and any k 2 N. Relations (2.4) imply that two block matrices Φ and Φ0 of this kind define isomorphic objects from E(M) if and only if Φ can be transformed into Φ0 by a sequence of the following transformations: Φe 7! Te Φe , where Te are invertible matrices and Ted⇤ = Ted+1 for all possible values of d, k; k

k

d⇤

d+1

Φf 7! Φf T f , where T f are invertible matrices and T fk = T fk n,s

= T f0 if n = 4(p − 1), then, moreover, Tem,s 0

for all possible values of d, k;

for all s 2 N (not for s = 1);

Φe 7! Uee0 Φe0 if e0 < e, where Uee0 is an arbitrary matrix of the appropriate size; 0

Φf 7! Φf U f

0f

if f 0 > f, where U f

0f

is an arbitrary matrix of the appropriate size.

These rules show that the classification of polyhedra in Snp actually coincides with the classification of representations from the bunch of chains X = {Ed , Fd , <, ⇠ | n  d  m} [cf. [6] or [7] (Appendix B)], where

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the relation ⇠ is defined by the exclusive rules: d+1 ed⇤ k ⇠ ek

and

fkd⇤ ⇠ fkd+1

for

n < d  2(n − p),

k 2 N,

and if n = 4(p − 1),then em,s ⇠ f0n,s 0

for

s2N

(but not for

s = 1).

Thus, the description of indecomposable representations given in [6, 7] implies a description of indecomposable polyhedra from Snp . We now recall the required combinatorics. We write e − f and f − e if e 2 Ed and f 2 Fd (with the same d) and set |X| = E [ F. Definition 4.2. such that

(1) A word is a sequence w = x1 r1 x2 r2 . . . xl−1 rl−1 xl , where xi 2 |X|, ri 2 {−, ⇠}

a) ri 6= ri+1 for all 1  i < l − 1;

b) xi ri xi+1 , 1  i < l, according to the definitions of the relations ⇠ and − presented above; c) if r1 = − (rl−1 = −), then x1 ⌧ y for all y 2 |X| (respectively, xl ⌧ y for all y 2 |X|).

We say that l is the length of the word w and write l = ln w. (2) For a word w, as specified above, we denote E(w) = { i | 1  i  l, xi 2 E }

and

F(w) = {i | 1  i  l, xi 2 F}.

(3) The inverse word w⇤ of the word w is the word xl rl−1 xl−1 . . . r2 x2 r1 x1 . (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 2 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+1 rk+1 . . . rk−1 xk (obviously, it is sufficient to consider 0  k < l ). (6) A cycle w is said to be nonperiodic if w 6= w[k] for 0 < k < l. (7) For a cycle w and an integer 0 < k < l, we denote by ⌫(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 2 |X|, there are no symmetric words and symmetric cycles in a sense of [7] (Appendix B). The words and cycles are associated with indecomposable representations of the bunch of chains X called strings and bands. We now describe the corresponding matrices Φ (recall that we have already excluded the part Φem fn ). Definition 4.3.

(1) If w is a word, then 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 located at the sites (i, i + 1) if ri = − and i 2 E(w) and (i + 1, i) if ri = − and xi 2 F(w); they are equal to 1.

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We denote the corresponding polyhedron by A(w); it is called a string polyhedron whenever it does not coincide with a sphere, a Moore polyhedron or a Chang polyhedron.1 (2) If w is a nonperiodic cycle, z 2 N, and ⇡ 6= t is a unital irreducible polynomial of degree v from (Z/p)[t], then the band matrix Φ(w, z, ⇡) is a block matrix in which all blocks are zv ⇥ zv in size; this matrix i 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 located at the sites (i, i + 1) if ri = − and i 2 E(w) and (i + 1, i) if ri = − and i 2 F(w) (note that the case i = l is also possible here); these nonzero blocks are equal to Izv (the zv ⇥ zv identity matrix), except the block located at the site (l1) (if l 2 E(w)) or (1l) (if l 2 F(w)); this block is the Frobenius matrix with the characteristic polynomial ⇡ v . If ⇡ = t − c is linear, then we replace the Frobenius matrix by the z ⇥ z Jordan block with the eigenvalue c.

We denote the corresponding polyhedron by A(w, z, ⇡) and call it a band polyhedron.2

By using these notions, we get the following description of polyhedra in the category Snp : Theorem 4.1. (1) All string and band polyhedra are indecomposable and every indecomposable polyn hedron from Sp , except spheres and Moore (or 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, ⇡) ' A(w⇤ , z, ⇡); A(w, z, ⇡) ' A(w[k] , z, ⇡ ⇤ ), where ⇡ ⇤ = ⇡ if ⌫(k, w) is even and ⇡ ⇤ (t) = tz ⇡(0)−1 ⇡(1/t) if ⌫(k, w) is odd.3 (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 way of writing of the words because, for every x 2 |X|, there is at most one element y 2 |X| such that x ⇠ y and, therefore, x − y is impossible. Hence, we can omit all symbols − (d−2)⇤ d−1 fl0 , then this means that and write x instead of x ⇠ y. Thus, if we write edk fld−1 ek0 (d−1)⇤

edk ⇠ ek

(d−2)⇤

− fld−1 ⇠ fl

(d−1)⇤

− ed−2 k 0 ⇠ ek 0

(d−2)⇤

− fld−1 ⇠ fl 0 0

.

It is possible to prove that one can find at most one place in a word w containing a fragment em,s ⇠ f n,s or ⇠ em,s 0 ; moreover, if this fragment is encountered, then w cannot be a cycle. f0n,s

1

The words consisting of one letter x correspond to spheres, the words of the form x ⇠ y correspond to Moore polyhedra, and the words with only one symbol ‘ − ’ correspond to Chang polyhedra; these words exhaust all possible exceptions. 2 Band polyhedra never coincide with spheres and Moore (or Chang) polyhedra. 3 v−1 If ⇡ = tv + a1 tv−1 + . . . + av−1 t + av , then ⇡ ⇤ = tv + a−1 + . . . + a1 t + 1). v (av−1 t

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Example 4.1. We now present 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 obtained for n = 6. They have three cells and are given by 7 6 6⇤ the words e6⇤ k f0 or e0 fl . The smallest band atoms have four cells. They are A(w0 , 1, t ⌥ 1), where 7 7 w0 = ek fl . Their gluing diagrams are as follows: •

11

•

• 3l

3l

•

10

•

9 ±1

8 •

7

•

3k

3k

•

6

•

•

(2) More complicated band atoms are A(w0 , 1, t2 + 1) and A(w0 , 2, t ⌥ 1). Their gluing diagrams are as follows: •

11

• 3l

•

10

•

•

3l

3l

•

•

3l

•

−1

9 ±1

±1

8 •

7 3k

6

• 3k

•

• 3k

•

• 3k

•

•

The attachments of cells ten come, respectively, from the Frobenius matrix ◆ ◆ ✓ of dimension ✓ nontrivial ±1 1 0 −1 . and the Jordan block 0 ±1 1 0

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(3) For the maximal value n = 8, the smallest atoms contain four cells. They are given by the words e80 f08,s f011 and have the following gluing diagrams: •

15 14 13 12 3s

• •

11 10 9 •

8

(4) The band atoms for n = 8 are quite complicated and cannot be “small.” Thus, as one of the smallest, 9⇤ 10⇤ 11 10 9 we can mention A(w, 1, t ⌥ 1), where w = e8⇤ k1 fl1 ek2 fl2 ek3 fl3 . The gluing diagram for this atom is as follows: •

15 14

•

13

•

• • •

12 11

•

10

•

9

•

8

•

(the powers of 3 near the vertical lines are omitted).

±1

• •

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(5) Finally, we present an example of an atom with exactly one cell of each dimension (we do not present the corresponding word because 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 atom in a sense of [15] (Chapter 14). One can also find the genera of p-primary polyhedra for 2p  n  4(p − 1). Namely, let ⇤(X) denote the ring Hos(X, X)/ tors(X). We say that the end x1 or xl of a word w is spherical if it of the form ed0 or f0d . Note that these letters can occur only at an end of the word because they are not related by ⇠ to any letter. It is easy to verify that ⇤(X) = 0 if X is a band polyhedron, while for a string polyhedron X = A(w), we have 8 0 if w has no spherical ends, > > > > < ⇤(X) = 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 where X = A(w) and both ends of the word w are spherical. In this 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. Thus, A contains Moore polyhedra Mkq (including S q = M0q ) and B contains the shifted Chang polyhedron C m = C00 [2p − 2]. Let Nk = Hosp (Mkq , C m ).

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Applying Hosp (Mkq , ) to the cofiber sequence 0 ! S m−1 ! S n ! C m ! S m ! S n+1 , we get an exact sequence µ

λ

0 ! Z/p − ! Nk − ! Z/p ! 0. Thus, #(Nk ) = p2 . On the other hand, applying Hosp ( , C) to the cofiber sequence (Edk ) of Section 2, we get an exact sequence pk

⌘

N0 −! N0 − ! Nk ! 0. Therefore the map ⌘ is an isomorphism. Setting k = 1, we see that pN0 = 0 and, 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 Im λ and by µk a generator of Nk such that µ(µk ) 6= 0. q q Similar observations show that the generator of the cyclic group Mqq kl = Hosp (Ml , Mk ) induces an isomorphism Nk ! Nl for k ≥ l > 0 and the zero map for 0 < k < l. On the other hand, � diagram (3.1) implies that � an element (a, b) of the ring ∆ = Hosp (C, C) acts on Nk as the multiplication by a recall that a ⌘ b (mod p) . Therefore, a map ' : A ! B, where A is a wedge of Moore polyhedra Mkq and B is a wedge of Chang polyhedra C m can be regarded as a block matrix Φ = (Φik )k2N[{0} , i=1,2

where all blocks are with coefficients from Z/p and both horizontal stripes Φ1 and Φ2 have the same number of rows. Namely, Φ1k consists of the coefficients of λk and Φ2k consists of the coefficients of µk . Two matrices of this sort define isomorphic objects from E(M) if and only if one of them can be transformed into the other by the following sequence of transformations: Φ1 7! T Φ1 and Φ2 7! T Φ2 with the same invertible matrix T ; Φk 7! Φk T k for some invertible matrix T k ; Φk 7! Φk + Φl Ulk 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 and, hence, the problem of classification of representations of any finitely generated algebra over the field Z/p (cf. [9], Section 5). Namely, we consider the case where the matrix Φ = Φ(F, G) has the form 2

I

0

6 6 0 I 6 6 6 F I 4 G 0

0

3

7 0 7 7 7. 0 7 5 I

Here I is the unit matrix of a certain size, F and G are arbitrary square matrices of the same size, and the lines show the subdivision of Φ into blocks Φik (there are only two vertical stripes). One easily see that Φ(F, G) and Φ(F 0 , G0 ) specify isomorphic objects if and only if there is an invertible matrix T such that F 0 = T F T −1 and G0 = T GT −1 . Thus, we arrive at 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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