Matrix problems, triangulated categories and stable homotopy types

MA TRIX PR OBLEMS, TRIANGULA TED CA TEGORIES AND ST A B LE HOMOTOPY TYPE S YURIY A. DROZD Abstract. W e show how the ma tr ix pro blems can b e used in studying triangulated categories . Then w e apply the general tec h- nique to the cla ssification of s table homotopy t yp e s of p olyhedra , find o ut the “representation t ypes” of such problems a nd give a description of stable homotopy types in finite and tame cas e s . Contents 1. Matrix problems a rising in triang ulated categories 2 2. Stable homotop y category 4 3. Discrete case : Whitehead–Chang Theorem 9 4. T ame case: Baues–He nnes Theorem 14 5. T orsion free polyhedra. Finite case 18 6. T orsion free polyhedra. T ame case 23 7. Wild cases 28 References 31 The tec hniq ue of matrix problems, esp ecially , of bimo dule cat ego ries, has pro v ed their efficiency in lots of problems from represen tation the- ory , algebraic geometry , group theory and other domains of mo dern algebra. During last y ears, mainly due to the w orks of Baues, Henn, Hennes, and the author, it has found new applications in algebraic top ology , na mely , in studying stable homotopy classes of p olyhedra (see [1 9], [8], [4]–[7], [1 6]). In the surve y [15] the author has pick ed out the bac kground of this appro ac h, whic h is based on the trianguled structure of thew stable homotop y category . In this pap er w e sho w t ha t the same metho d can b e used in general situation, when w e construct sub categories of a tria ngulated category from simpler ones (see Sec- tion 1). Then w e summarize what can be done using this techniq ue for 1991 Mathematics Subje ct Classific ation. 5 5P15, 55P 42, 18 E30, 1 6G60. Key wor ds and phr ases. stable homotopy t yp es, po lyhedra, triangulated cate- gories, matrix pro blems, representation type. This paper summarizes the investigations mainly made during the sta y of the author at the Max-Plank-Insitut f¨ ur Mathema tik in Bonn. The author was also partially supp orted by the INT AS Gra n t 06-100 0017- 9093. 1 2 YURIY A. DROZD the classification problem o f stable homotop y classes. Namely , w e con- sider t he sub cat ego ries S n of the stable homotop y catego ry consisting of p olyhedra ha ving only cells of n consecutiv e dimensions. W e classify p olyhedra f rom S n for n ≤ 4 and sho w that for n > 4 this problem b e- comes wild in the sense o f the represen tation theory of alg ebras. Then w e consider t he sub categories T n of S n consisting o f p olyhedra with no torsion in on t egral homologies. This time w e classify p o lyhedra from T n for n ≤ 7 a nd show that for n > 7 their classific ation is also a wild problem. In some sense, t hese results are “final,” though we are sure that this tec hnique will b e useful for some other problems o f algebraic top ology as well a s for studying other triangulated categories. Since the tec hnical details of calculations are sometimes rather cum- b ersome and can b e found in the previous pap ers, w e usually omit them, just outlining the ideas. 1. Ma trix pro b lems arising in triangula ted ca tegorie s Let C b e a triangulated category with the shift A 7→ S A , A and B b e tw o fully additiv e (but usually not triangulated) sub categories of C . W e denote by A † B the full sub cat ego ry of C consisting of all ob jects C arising in triangles (1.1) A a − → B b − → C c − → S A with A ∈ A , B ∈ B . W e also denote b y I the ideal of the category C consisting of all morphisms γ : C → C ′ that fa ctorizes b oth through B and through S A , i.e. suc h that γ = γ ′ α = γ ′′ β , where α : C → S A, β : C → B , where A ∈ A , B ∈ B . On the other hand, w e consider the A - B -bimo dule B C A , whic h is the restriction o f t he regular C -bimo dule C ( A, B ) for A ∈ A , B ∈ B . W e often o mit subscripts and denote this bimo dule b y C if it cannot lead to misunderstanding. Recall that the bimo dule c ate gory Bim ( B C A ) has S A ∈ A B ∈ B C ( A, B ) as the set of ob jects, while the set of mor phisms Bim ( a, a ′ ), where a : A → B , a ′ : A ′ → B ′ , is defined as { ( α, β ) | α : A → A ′ , β : B → B ′ , β a = a ′ α } . W e denote b y J the ideal of Bim ( B C A ) consisting o f all morphisms ( α, β ) : a → a ′ suc h t ha t α factors through a and β f actors through a ′ . W e define a functor F : Bim ( B C A ) → ( A † B ) / I as follows . F or ev ery morphism a : A → B , c ho ose a triangle lik e (1.1) and set C = F a . If a ′ : A ′ → B ′ , C ′ = F a ′ and ( α, β ) ∈ Bim ( a, a ′ ), there is γ : C → C ′ suc h that the diagra m (1.2) A a − − − → B b − − − → C c − − − → S A S a − − − → S B α   y β   y γ   y   y S α   y S β A ′ a ′ − − − → B ′ b ′ − − − → C ′ c ′ − − − → S A ′ S a ′ − − − → S B ′ MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 3 comm utes. Set F ( α, β ) = γ mo d I . W e m ust ch ec k t ha t the latter definition is consisten t. Indeed, if γ ′ : C → C ′ is ano ther mor phism making diag ram (1.2) commutativ e, g = γ − γ ′ , then g b = c ′ g = 0, therefore t here are f : S A → C ′ and h : C → B ′ suc h that g = cf = b ′ h , i.e. g ∈ I . Thu s F is w ell-defined. Supp ose now that C ( B , S A ) = 0 for all A ∈ A , B ∈ B . In this situation w e define a functor G : A † B → Bim ( B C A ) / J as f o llo ws. Let C ∈ A † B . Cho ose one triangle lik e (1.1) a nd set a = GC . If GC ′ = a ′ , i.e. C ′ o ccur in t he triangle A ′ a ′ − → B ′ b ′ − → C ′ c ′ − → S A ′ with A ′ ∈ A , B ′ ∈ B , and γ : C → C ′ , then c ′ γ b = 0, hence γ b = b ′ β for some β : B → B ′ . Cho ose one of suc h triangles Since B b − → C c − → S A − S a − − → S B and B ′ b − → ′ C ′ c − → ′ S A ′ − S a ′ − − − → S B ′ are also tria ngles, t here is a morphism α : A → A ′ that make s the dia- gram (1.2 ) comm uta tiv e, thus ( α , β ) ∈ Bim ( a, a ′ ). Set Gγ = ( α , β ) mo d J . If ( α ′ , β ′ ) is another pair making (1.2) commutativ e, t hen ( β − β ′ ) b ′ = 0, hence β − β ′ = a ′ f for some f : B → A ′ ; in the same wa y S α − S α ′ = g ( S a ), i.e. α − α ′ = g ′ a for some g : S B → S A ′ and g ′ : B → A ′ suc h that S g ′ = g . Therefore ( α − α ′ , β − β ′ ) ∈ J , so the functor G is w ell-defined. Theorem 1.1. Supp ose that C ( B , S A ) = 0 for al l A ∈ A , B ∈ B . Then the functors F , G c onstructe d ab ove ind uc e quasi- i n verse func- tors ¯ F : Bim ( B C A ) / J → ( A † B ) / I and ¯ G : ( A † B ) / I → Bim ( B C A ) / J . Thus ( A † B ) / I → Bim ( B C A ) / J . Mor e over, I 2 = 0 , ther efor e, the natur al functor Π : ( A † B ) → ( A † B ) / I is an epivalenc e. Recall that an epivalenc e is a functor E : C 1 → C 2 , whic h is • ful l , i.e. all induced maps C 1 ( X , Y ) → C 2 ( E X , E Y ) a r e surjec- tiv e; • dense , i.e . eve ry ob ject from C 2 is isomorphic to E X for some X ∈ C 1 ; • c onservative , i.e. f ∈ C 1 ( X , Y ) is in v ertible if and only if so is E f ∈ C 2 ( E X , E Y ). (In [2] suc h functors are called d e te cting .) No te that then also • X ≃ Y in C 1 if a nd only if E X ≃ E Y in C 2 ; • if C 1 and C 2 are additiv e, then an ob ject X ∈ C 1 is indecom- p osable (in to a non trivial direct sum) if and only if so is E X . Pr o of. One immediately sees that F ( J ) = 0 and G ( I ) = 0, hence ¯ F and ¯ G are w ell-defined. Moreov er, w e hav e already seen that, giv en 4 YURIY A. DROZD ( α, β ), the morphism γ is defined up to a summand from I , and g iv en γ , the pair ( α, β ) is defined up to a summand from J . It obv io usly implies t hat ¯ F ¯ G ≃ Id nol imits and ¯ G ¯ F ≃ Id nol imits . If γ : C → C ′ and γ ′ : C ′ → C ′′ are from I , then γ = g f for some f : C → B and g : B → C ′ , where B ∈ B , while γ ′ = g ′ f ′ for some f ′ : C ′ → S A and g ′ : A → C ′′ , where A ∈ A . Then γ ′ γ = g ′ f ′ g f = 0, since f ′ g ∈ C ( B , S A ). Th us I 2 = 0 and, therefore, Π is an epiv a lence.  Corollary 1.2. Under c onditions of The or em 1.1, let V b e a subbi- mo dule of B C A such that f 1 af 2 = 0 whenev e r a ∈ V , f i ∈ C ( B i , A i ) with A i ∈ A , B i ∈ B ( i = 1 , 2) . Denote by A † V B the ful l sub c a t- e go ry of A † B c onsisting of al l o b je cts C ari s i ng in triangles (1.1) with a ∈ V , I V = I ∩ ( A † V B ) , J V = J ∩ Bim ( V ) . Then the functor F an d G c onstructe d ab ove induc e quasi-inverse functors ¯ F : Bim ( V ) / J → ( A † V B ) / I V and ¯ G : ( A † V B ) / I → Bim ( V ) / J V . Thus ( A † V B ) / I ≃ Bim ( V ) / J V . Mor e ov e r, I 2 V = 0 and J 2 V = 0 , ther efor e, the natur al functors ( A † V B ) → ( A † V B ) / I V and Bim ( V ) → Bim ( V ) / J 2 V ar e epivalenc es. In p articular, t h e r e is a one- to-one c orr es p ond enc e s b etwe en isomorphism classes of ob je cts a n d of inde c om p osa b le ob j e cts fr om A † V B an d Bim ( V ) . 2. St able homotopy ca teg or y In this pap er the w o rd “ p olyhe dr on ” is used as a synon ym f or“ finite c el l (or CW) c omp l e x ”. W e denote b y Hot the catego r y of punc- tured top olo gical spaces with homotopy classes of con tinu ous maps as morphisms and by CW its f ull sub category consisting of p olyhe- dra. W e denote b y C X the c one over the sp ac e X , i.e. the factor space X × I / X × 1, I = [0 , 1] b eing the unit inte rv al. F or a ma p f : X → Y w e denote b y C f the c one of this map , i.e. the factor space ( Y ⊔ C X ) / ∼ , where the equiv alence relation ∼ is given by the rule f ( x ) ∼ ( x, 0). Let also S X b e the susp en s ion of X , i.e. t he f actor space C X/ ( X × 0). This operatio n induces a functor S : Hot → Hot . Note that f o r ev ery X the space S X is an H -c o gr oup and the n -fold suspension S n X is a c o m mutative H - c o gr oup for n ≥ 2 [23, 2.21 – 2.26]. Therefore, Hot ( S n X , Y ) is a group, comm uta tiv e for n ≥ 2. The natura l maps Hot ( S n X , S n Y ) → Hot ( S n +1 X , S n +1 Y ) are group homomorphisms. Set Hos ( X, Y ) = lim − → n Hot ( S n X , S n Y ) . It is a group called the group of stable maps from X to Y . Th us w e get the stabl e homotopy c ate gory Hos and its full sub category S consisting of p olyhedra. W e also denote by CF and T respectiv ely the full subcategories of CW and o f S consisting of torsion fr e e p olyhe dr a X , i.e. suc h that all integral homology groups H k ( X ) = H k ( X , Z ) are MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 5 torsion fr ee. The g roups Hos ( S n , X ) are called the stable ho motopy gr oups o f the space X and denoted b y π S n ( X ). The category Hos is additiv e, with the b o uquet (or we dge ) X ∨ Y pla ying the role of direct sum. Moreov er, Hos is ful ly additive , i.e. ev- ery idemp o t ent in it splits [12, Theorem 4.8]. The susp ension induces a functor, whic h w e also denote b y S : Hos → Hos . Obviously , it is fully faithful. Th us w e can “ supplemen t” it so that S b ecomes an equiv a- lence. T o do it, w e consider formal “imag inary spaces” S n X with n < 0 setting, for n < 0 or m < 0, Hos ( S n X , S m Y ) = Hos ( S n + k X , S m + k Y ), where k = − min( n, m ). Then w e consider formal b ouquets W r i =1 X i , where eac h X i is either a “real” o r an “imagina r y” space, and de- fine Hos ( W s j = 1 Y j , W r i =1 X i ) as the set of r × s matrices ( f ij ) with f ij ∈ Hos ( Y j , X i ) (see [1 2] f o r details). As a resu lt w e get the category (also denoted b y Hos ), where S is an auto-equiv alence. In fact, the new category is a triangulate d category . The triangles in it are the c ofibr ation se quenc es , i.e. those isomorphic to the c one se q uenc es X f − → Y g − → C f h − → S X , where g is the natural em b edding Y → C f and h is the natural sur- jection C f → S X ≃ C f / Y [22]. Note that in the stable category Hos they coincide with the fibr ation se quenc es [12], though w e do not use this fact. W e denote b y CW k n the full subcategor y of CW consis ting of ( n − 1)- connected cell complexes of dimension at most n + k . If X ∈ CW k n , o ne can suppo se that it s ( n − 1)-th sk eleton X n − 1 (the “( n − 1)- dimensional part” of X ) consist of a unique p oint a nd X has no cells o f dimensions greater than n + k . F o llowing Baues, we describe suc h a cell complex using it s gluing (or attachment ) diagr am , whic h lo oks lik e (f o r n = 7 , k = 6) (2.1) 13 • 6                 • 2               12 • 8 * * * * * * * * * * * * * * * * •               11 • 2 * * * * * * * * * * * * * * * * • 1 * * * * * * * * * * * * * * * * • 3                 10 • • 9 • • • 8 • 7 • • • In this diagr a m eac h bullet on the leve l m corresp onds to an m -dimensional cell, i.e. to a ball B m glued to the ( m − 1)-dimensional sk eleton X m − 1 b y a map of its b oundar y f : S m − 1 → X m − 1 . The lines betw een this bullet and the low er o nes describ e the nonzero compo nents of the map f . If there ar e more than one nonzero map b etw een S m − 1 and a smaller 6 YURIY A. DROZD S l ( l < m ), these lines carry some marks precising the corresponding maps. Especially , in our example the groups Hos ( S l +3 , S l ) a re cyclic o f order 24, so we put t he marks that sho w, whic h m ultiple of the genera- tor is used for this gluing. There ar e no marks on other lines, sinc e the groups Hos ( S l +2 , S l ) a re of order 2, so only ha ve one nonzero elemen t . Ev ery p olyhedron fro m S decomp oses in t o a direct sum of indecom- p osable ones. Note that suc h a decomp osition is far from b eing unique (see [1 2, 4.2] for examples). Neve r t heless, a description of indecomp os- able p olyhedra in S can b e a go o d first step to w ards the classification problem. Moreo v er, if the endomorphism ring Es ( Y ) = Hos ( Y , Y ) is lo c al and Y ∨ Y ′ ≃ L i Y i , there is an index i su ch that Y i ≃ Y ∨ Y ′′ [1, Lemma I.3.5] Hence, in a ll decomp o sitions of a p olyhedron X in to b ouquets of indecomp osables the multiplicit y o f Y is the same. An- other approac h gives the notion of c o ngruenc e . Namely , we sa y that t w o p olyhedra X , Y are c ongruent if there is a p o lyhedron Z suc h that X ∨ Z ≃ Y ∨ Z . One can sho w, fo llo wing [2 1] or [13], that an equiv alen t condition is that the images of X and Y in all lo c aliza- tions S p of the stable homo t o p y category are isomorphic. Here S p ( p is a prime in teger) is the category whose ob jects are p olyhedra, but Hos p ( X , Y ) = Hos ( X , Y ) ⊗ Z p , where Z p is the ring of p -adique in t e- gers. (The same notion is obtained if w e r eplace Z p b y the subring { a/b | a, b ∈ Z , p ∤ b } of the r a tional num b ers.) W e call the classes of congruence g ener a , lik e they do in the t heory of integral represe n- tations. Though genera satisfy the cancellation prop ert y (in fact, b y definition), their decomposition into b ouquets of indecomp osable is not unique to o (se e the first of the cited examples fro m [12]). Recall that due to the G eneralized F reuden thal Theorem [12, The- orem 1 .21] there is no nee d to go up to infinit y in defining Hos ( X, Y ) if w e deal with p olyhedra. Namely , if Y is ( n − 1)-connected and dim X ≤ m , then the map Hot ( X , Y ) → Hot ( S X , S Y ) is bijectiv e if m < 2 n − 1 and surjectiv e if m = 2 n − 1. It implies that the map Hot ( S k X , S k Y ) → Hos ( X, Y ) is bij ective for k > m − 2 n + 1 and su r- jectiv e for k = m − 2 n + 1. In particular, if Y is ( n − 1)- connected, π S m ( Y ) ≃ π 2( m − n +1) ( S m − n +2 Y ). Moreo v er, on the sub category of Hot consisting of simply connected spaces the suspension functor is conser- v ative. Therefore, t he induced functor CW k n → CW k n +1 is an equiv alence for n > k + 1 and an epiv alence f or n = k + 1. Denote b y S n the image in S of the category CW n − 1 n . Th e p olyhedra from S n can only ha v e cells on n consecutiv e lev els (from n -th up to (2 n − 1)-th) and every p olyhedra ha ving cells on n consecutiv e lev els is isomorphic in S to S m X for some integer m and some X ∈ S n . W e a lso denote b y T n the full subcategory of S n consisting of to r sion free p olyhedra. Definition 2.1 (cf. [3]) . An atom is an indecomp osable ob ject A from S n , whic h do es not b elong to S ( S n − 1 ) ∪ S 2 ( S n − 1 ). (In other words, an y p olyhedron isomorphic to A in S m ust ha v e cells of dimensions n MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 7 and 2 n − 1.) If A is an atom, all p o lyhedra of the sort S m A are called susp en de d atoms . This definition immediately implies that ev ery p olyhedron is isomor- phic in S to a bouquet of susp ended atoms, though, as w e ha v e men- tioned, suc h a decomp osition is not unique. Note that, unlik e Baues, w e consider S 1 as an atom (a unique ato m in S 1 ), hence all spheres are considered a s suspended atoms. Note also that this definition implies that all atoms are of o dd dimensions: an atom from S n is of dimension 2 n − 1. T o clarify the structure of S n w e use the tec hnique from Section 1. Namely , c ho o se an in teger m suc h that 0 ≤ m < n − 1 and set A = A n,m = S 2 m +1 S n − m − 1 , B = B n,m = S n − m − 1 S m +1 , S n,m = B n,m S A n,m , I n,m = { f : X → Y | X , Y ∈ S n , f factors b oth through B a nd through S A } , J n,m = { ( α , β ) ∈ Bim ( a, a ′ ) | a, a ′ ∈ S n,m , α factors through a and β factors through a ′ } . (2.2) Then p olyhedra from A only hav e cells in dimens io ns from n + m up to n − m − 2, while those from B only hav e cell in dimensions fro m n up to n + m . If C ∈ S n , its ( n + m )-t h sk eleton B belongs to B , while the factor space C /B b elong s to S A , i.e. C /B ≃ S A, A ∈ A . Then C ∈ A † B , since A → B → C → C /B ≃ A is a cofibration sequence. On the other hand, an y o b ject from A † B obvious ly b elongs to S n . So we hav e pro v ed Theorem 2.2. S n ≃ A n,m † B n,m . Thus S n / I n,m ≃ Bim ( S n,m ) / J n,m . Mor e over, I 2 n,m = 0 . T o conside r torsion free p olyhedra, w e set A 0 = A 0 n,m = S 2 m +1 T n − m − 1 , B 0 = B 0 n,m = S n − m − 1 T m +1 , S 0 = S 0 n,m = { a ∈ S n,m | H n + m ( a ) = 0 } , I 0 n,m = I n,m ∩ ( A 0 † S 0 B 0 ) , J 0 n,m = J n,m ∩ B im ( S 0 ) . (2.3) T o get an analogue of The o r em 2.2 w e need the fo llo wing lemma. Lemma 2.3. L e t f ∈ Hos ( A, B ) , wher e A and B ar e torsion fr e e p ol yhe dr a, A is ( m − 1) -c on n e cte d, dim B ≤ m a nd C f is also torsion fr e e. Ther e ar e d e c omp ositions A ≃ C ⊕ A ′ , B ≃ C ⊕ B ′ such that, with 8 YURIY A. DROZD r esp e ct to this de c omp osition, f =  Id nol imits 0 0 g  with H m ( g ) = 0 and C f ≃ C g . Pr o of. Note first that if A = kS m , B = lS m are b ouquets o f m - dimensional spheres, then H m ( A ) = m Z , H m ( B ) = l Z , and t he natura l map Hos ( A, B ) → Hom nol imits (H m ( A ) , H m ( B )) is an isomorphism. In pa r ticular, ev ery decomp osition of H m ( A ) arises from a decomp o- sition of A , and the same is true for B . In this case H m ( f ), or, the same, f is actually an inte g er matrix a nd t here are decomp ositions A ≃ C ⊕ A ′ , B ≃ C ⊕ B ′ (all sum ma nds are, of course, also b ouquets of spheres) suc h that, with resp ect to them, f =  Id nol imits 0 0 d  , where d : A ′ → B ′ can b e prese nted b y diagonal matrix without unit comp onen ts. In general case, the calculation of homologies of cell spaces fro m [23 , Chapter 10 ] shows that the em b edding α : A m → A induce s a surjection H m ( A m ) → H m ( A ), while the surjection β : B → ˜ B = B /B m − 1 induces an embedding H m ( B ) → H m ( ˜ B ) with torsion f r ee cok ernel. Therefore, there ar e decompositions A m ≃ A 1 ⊕ A 0 , ˜ B ≃ B 1 ⊕ B 0 suc h that the restriction of H m ( α ) on to A 1 is an isomorphism, and that on to A 0 is 0, while H m ( f ) induces an isomorphism H m ( B ) → im nol imits H m ( β ) = H m ( B 1 ). Denote b y α 1 : A 1 → A and β 1 : B → B 1 the corre- sp onding compo nen ts of α and β . As ab o v e, there are decompo sitions A 1 ≃ C ⊕ A 0 , B 1 ≃ C ⊕ B 0 suc h that, with respect to them, the mor- phism β 1 f α 1 =  Id nol imits 0 0 d  , where d can b e presen ted b y diago- nal matrix without unit comp onen ts. D enote by ι : C → A 1 the natural em b edding (presen ted by the matrix  Id nol imits 0  ) a nd b y π : B 1 → C the natural pro jection (presen ted b y the matrix  Id nol imits 0  ). Then π β 1 f α 1 ι = Id nol imits , so B ≃ C ⊕ B ′ , A ≃ C ⊕ A ′ , so t ha t, with resp ect to these decomp ositions, f =  Id nol imits 0 0 g  . Then C f ≃ C g a nd d = β 0 g α 0 , where α 0 : A 0 → A ′ and β 0 : B ′ → B 0 . Note that α 0 and β 0 also induce isomorphisms of the m -th homol- ogy groups, so Coke r nol imits H m ( g ) ≃ Coke r nol imits H m ( d ). Since this cok ernel em b eds in H m ( C g ), it is to rsion free. Therefore, d = 0, whence H m ( g ) = 0.  Theorem 2.4. T n ≃ A 0 n,m † S 0 B 0 n,m . Thus T n / I 0 n,m ≃ Bim ( S 0 n,m ) / J 0 n,m . Mor e over, ( I 0 n,m ) 2 = ( J 0 n,m ) 2 = 0 , so this e quivalenc e induc es one- to-one c orr es p ond enc e s b etwe en isomorphism classes of ob je cts a n d of inde c om p osa b le ob j e cts in T n and in Bim ( S 0 n,m ) . MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 9 Pr o of. Let C ∈ T n , B = C n + m , S A ≃ C /B . The triangle (2.4) A a − → B b − → C c − → S A giv es rise to the exact seque nce of homologies · · · → H k ( A ) H k ( a ) − − − → H k ( B ) H k ( b ) − − − → H k ( C ) H k ( c ) − − − → H k ( S A ) ≃ ≃ H k − 1 ( A ) H k − 1 ( a ) − − − − → H k − 1 ( B ) H k − 1 ( b ) − − − − → H k − 1 ( C ) → . . . If k < n + m , then H k ( A ) = H k − 1 ( A ) = 0, so H k ( B ) ≃ H k ( C ) is to rsion free. If k > n + m , w e get in the same w a y that H k ( A ) ≃ H k +1 ( C ) is also torsion free. Let no w k = n + m , then we get the ex a ct sequenc e 0 → H n + m +1 ( C ) → H n + m ( A ) H n + m ( a ) − − − − − → H n + m ( B ) → H n + m ( C ) → 0 . Note that H n + m ( B ) is a lw a ys torsion free, since B contains no ce lls of dimensions bigger than n + m , hence B ∈ T . Th erefor e, H n + m ( A ) is torsion free to o, so A ∈ T . Moreo ve r, Cok er nol imits H n + m ( a ) is also torsion free. As b oth H n + m ( A ) and H n + m ( B ) ar e f r ee, it means that H n + m ( A ) ≃ M ⊕ M ′ , H n + m ( B ) ≃ M ⊕ M ′′ so that H n + m ( a ) induces isomorphism M → M and is zero on M ′ . By Lemma 2 .3 , there are decomp ositions A ≃ A 0 ∨ A ′ , B ≃ A 0 ∨ B ′ suc h that, with resp ect to them, a =  Id nol imits 0 0 a ′  , where a ′ ∈ S 0 . Then C a ′ ≃ C a ≃ C , so C ∈ A 0 † S 0 B 0 . On the other hand, if C ∈ A 0 † S 0 B 0 , i.e. b elongs to a triangle (2.4) with A ∈ A 0 , B ∈ B 0 and H n + m ( a ) = 0, the exact sequence of homologies implies that C ∈ T n . T o prov e the remaining assertions, it is enough to show that uav = 0 for ev ery a ∈ S 0 ( A, B ), v : B ′ → A, u : B → A ′ , where A, A ′ ∈ A 0 , B , B ′ ∈ B 0 (see Corollary 1.2). Since H n + m ( a ) = 0, the induce d map A m + n → B /B m + n − 1 is zero. On the other hand, S ( B ′ , A/ A m + n ) = 0 = S (( B ′ ) m + n − 1 , A m + n ), so the map v : B ′ → A factors through a map B ′ / ( B ′ ) m + n − 1 → A m + n . Since the same holds for u , it implies that uav = 0.  W e shall a lso use the follo wing ob vious lemma. Lemma 2.5. L et X ∈ S n , H i = H i ( X ) . If X is de c omp osable, ther e ar e de c omp ositions H i = H ′ i ⊕ H ′′ i and indic es j, k s uch that b oth H ′ j 6 = 0 and H ′′ k 6 = 0 . (Note a lso that H i ( X ) = 0 for i < n or i > 2 n − 1 .) 3. Discrete case: Whitehead–Chang Theorem W e apply now Theorem 2.2 to p olyhedra from S n for small n . First, w e recall some v alues of stable homotop y groups [20, Sec tio ns XI.15– 16]: • π S n +1 ( S n ) ≃ Z / 2, the generator b eing the (suspended) Hopf m ap η = 2 n − 1 h 2 , where h 2 is the Hopf fibration S 3 → S 2 ; 10 YU RIY A. DROZD • π S n +2 ( S n ) ≃ Z / 2, t he generator b eing the double Hopf map η 2 , i.e. the comp osition of Hopf maps S n +2 → S n +1 → S n ; • π S n +3 ( S n ) ≃ Z / 24, the generator b eing ν = S n − 4 h 4 , where h 4 is the Ho pf fibration S 7 → S 4 . Moreo v er, the comp osition η 3 : S n +3 → S n +2 → S n +1 → S n equals 12 ν . If n = 1, the only a tom in S 1 is S 1 , a nd ev ery p olyhedron is a b ouquet of sev eral copies of S 1 . If n = 2, S 2 = A 2 , 0 † B 2 , 0 , and A 2 , 0 = B 2 , 0 = S S 1 . Th us ev ery po lyhedron C from S 2 is isomorphic to the cone of a map a : k S 2 → l S 2 .Since Hos ( S 2 , S 2 ) = Z , the map a can b e considered as a matrix ( a ij ) ∈ Mat nol imits ( l × k , Z ). If a ′ is another ob ject from C 2 , 1 , also considered a s a matrix from Mat nol imits ( l ′ × k ′ , Z ), a morphism a → a ′ in Bim ( C 2 , 1 ) is giv en by a pair of matrices α ∈ Mat nol imits ( k ′ × k , Z ), β ∈ Mat nol imits ( l ′ × l , Z ) suc h that a ′ α = β a . Esp ecially , this morphism is an isomorphism if and only if b oth α and β are inv ertible. So the w ell-kno wn Smith Theorem implies that ev ery o b ject a ∈ C 2 , 1 is isomorphic to o ne presen ted by a diagonal matrix diag nol imits ( q 1 , q 2 , . . . , q r ). Hence, ev ery p olyhedron from S 2 is isomorphic to a b ouquet of cones W i C q i , where w e iden tify an integer q with the corresp onding map S 2 → S 2 . Moreo ve r, if q = uv , where gcd( u, v ) = 1, then C q ≃ C  1 0 0 q  ≃ C  u 0 0 v  ≃ C u ∨ C v . Therefore, C q can only b e indecompo sable if q = p s , where p is prime. On the other hand, the exact sequenc e of homologies arising from the triangle (3.1) S 2 q − → S 2 − → M 3 ( q ) − → S 3 that H 2 ( C q ) ≃ Z /q and H 3 ( C q ) = 0. Hence, Lemma 2.5 implies that C q is indecomp osable. Therefore, the a toms in S 2 are just C q for q = p s with a prime p . These atoms are denoted b y M 3 ( q ) and t heir suspensions S k M 3 ( q ) b y M k +3 ( q ). The atoms and susp ended atoms M d ( q ) are called Mo or e sp a c es 1 [12]. W e also write M d s instead of M d (2 s ) (t hese atoms play a special ro le later). W e can calculate the groups Hos ( M 3 ( q ) , M 3 ( q ′ )). Since π S 3 ( S 2 ) ≃ Z / 2 [20, Theorem 15.1 ], the exact sequences for t he functor Hos ar ising from the triangles (3.1) for q and q ′ imply that Hos ( S 2 , M 3 ( q )) ≃ Hos ( M 3 ( q ) , S 3 ) ≃ Z /q , Hos ( S 3 , M 3 ( q )) ≃ Hos ( M 3 ( q ) , S 2 ) ≃ ( Z / 2 if q is ev en , 0 if q is o dd , Hos ( M 3 ( q ) , M 3 ( q ′ )) ≃ Z / ( q , q ′ ) if q or q ′ is odd , 1 In [20, Section XI.10] they are denoted by P d q and called pseudo-pr oje ctive sp ac es . MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 11 and there is an ex a ct sequenc e 0 → Z / 2 → Hos ( M 3 s , M 3 r ) → Z / 2 m → 0 , where m = min( r , s ) . (3.2) Note that the endomorphism rings Es ( M 3 ( q )) are finite, hence, lo cal. These considerations immediately imply the description of p olyhedra from S 2 . Theorem 3.1. Every p o l yhe dr on fr om S 2 uniquely (up to p ermutation of summa n ds) de c om p ose s into a b ouquet of spher es S 2 , S 3 and Mo or e atoms M 3 ( q ) . W e also need the follo wing fact. Prop osition 3.2. π S 4 ( M 3 ( q )) ≃      0 if q is odd , Z / 4 if q = 2 , Z / 2 ⊕ Z / 2 if q = 2 s , s > 1 . Pr o of. Recall that π S 4 ( S 3 ) ≃ π S 4 ( S 2 ) ≃ Z / 2 [20 , Theorems 15.1, 15.2 ]. Therefore, the exact sequence for π S 4 arising from (3.1 ) sho ws that π S 4 ( M 3 ( q )) = 0 fo r q odd and, for q = 2 s , there is an exact sequence 0 → Z / 2 → π S 4 ( M 3 s ) → Z / 2 → 0 . Note that π S 4 ( M 3 1 ) ≃ π 6 ( M 5 1 ), so [20, L emma 10.2] implies that it em- b eds in to π 6 ( S 3 ) ≃ Z / 12 [20, Theorem 16.1]. Hence, π S 4 ( M 3 1 ) ≃ Z / 4. F or r > 1 consider t he comm utative dia gram of tria ngles (3.3) S 2 2 − − − → S 2 − − − → M 3 1 − − − → S 3 1   y 2 r − 1   y   y 1   y S 2 2 r − − − → S 2 − − − → M 3 s − − − → S 3 , It induces the comm utative diagram with exact ro ws 0 − − − → Z / 2 − − − → π S 4 ( M 3 1 ) − − − → Z / 2 − − − → 0 0   y   y   y 1 0 − − − → Z / 2 − − − → π S 4 ( M 3 s ) − − − → Z / 2 − − − → 0 , whic h sho ws tha t the second row is the pushdo wn of the first one along zero map, hence, it splits.  Prop osition 3.3. Hos ( M d s , M d r ) ≃ ( Z / 4 if r = s = 1 , Z / 2 ⊕ Z / 2 m otherwise, where m = min( r , s ) . 12 YU RIY A. DROZD Pr o of. Ob viously , w e ma y supp ose that m = 3. Since π S 4 ( M 3 1 ) ≃ Z / 4 is a module o v er the ring Hos ( M 3 1 , M 3 1 ), 2 Hos ( M 3 1 , M 3 1 ) 6 = 0, hence, Hos ( M 3 1 , M 3 1 ) ≃ Z / 4. On the other hand, applying the func- tor Hos ( , M 3 1 ) t o the diagram (3.3) with s > 1, w e get a comm utative diagram with exact row s 0 − − − → Z / 2 − − − → Hos ( M 3 1 , M 3 1 ) − − − → Z / 2 − − − → 0 1 x   x   x   0 0 − − − → Z / 2 − − − → Hos ( M 3 s , M 3 1 ) − − − → Z / 2 − − − → 0 . Th us its second ro w is the pull-back of the first one along the zero map, hence, it splits. The dual conside ra tion show s that the sequence (3 .2) for r > 1 can b e obtained as a pushdo wn of the sequence for r = 1, hence, it splits to o.  Note that the latter decomp o sition in this stateme nt is that o f gr oups . T aking into account the m ultiplication, it is con v enien t to presen t mor- phisms M d s → M d r as t r ia ngular matrices  a b 0 c  , w it h a ∈ Z / 2 r , b ∈ Z / 2 , c ∈ Z / 2 s , 2 s − m a ≡ 2 r − m c mo d 2 µ , where m = min ( s , r ) , µ = max( s, r ). The pro duct of morphisms corr esp ond then to the usual pro duct of matr ices, while the sum of mor phisms corresp ond to the usual sum of matrices, with the only exception, when s = r = 1: then w e m ust add matrices as follow s:  a b 0 a  +  a ′ b ′ 0 c ′  =  a + a ′ b + b ′ + aa ′ 0 a + a ′  . Let no w n = 3 , m = 1, then S 3 = A 3 , 1 † B 3 , 1 , where A 3 , 1 = S 3 S 1 and B 3 , 1 = S S 2 . Hence, p olyhedra from A 3 , 1 are just b ouquets of spheres S 4 , while those from B are b ouquets of spheres S 4 , S 3 and Mo ore spaces M 4 ( q ). F o r con v enience, we set M 4 0 = S 4 and M 4 ∞ = S 3 and order the set of indices b y the rule 1 < 2 < · · · < ∞ < 0. As w e ha ve seen, Hos ( S 4 , M 4 ( q )) = 0 fo r q odd, Hos ( S 4 , M 4 r ) = H r ≃ Z / 2 for r 6 = 0 and Hos ( S 4 , S 4 ) = H 0 ≃ Z . Therefore, a map a : A → B , where A ∈ A , B ∈ B can be presen ted as a blo c k matrix (3.4) a =       a 0 a ∞ . . . a 2 a 1       , where a s are matrices ov er H s . One easily sees that if η s is a generator of H s and β r s : M 4 s → M 4 r , then β r s η s = 0 if r > s , while for r ≤ s the MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 13 map β r s can be so c hosen that β r s η s = η r . Set H r s =      0 if r > s, Z / 2 if 0 6 = r ≤ s, Z if s = r = 0 . Therefor t w o matrices a, a ′ of the form (3.4) define isomorphic ob jects from Bim ( S 3 , 1 ) if a nd only if there is an inv ertible integral matrix α and a n inv ertible blo c k matrix β = β r s , where β r s is a matrix o ver H r s , suc h that a ′ = β aα − 1 . Then simple considerations show that ev ery ob ject from Bim ( S 3 , 1 ) decomp oses in to a direct sum of ob jects give n b y the 1 × 1 matrices q ∈ H 0 , η r ∈ H r , r 6 = 0 and  2 s η r  ∈ H 0 ⊕ H r , r 6 = 0 , s > 0 . The first case corresp ond t o the Mo o re space M 5 ( q ), while the second and the third cases define new p olyhedra, resp ectiv ely , C 5 ( η ), C 5 (2 r η ), C 5 ( η 2 s ) a nd C 5 (2 r η 2 s ), giv en b y the gluing dia g rams 5 • • } } } } } } } } } } } } • } } } } } } } } } } } } • w w w w w w w w w w w w w w 4 • • • • 3 • • • • C 5 ( η ) C 5 (2 r η ) C 5 ( η 2 s ) C 5 (2 r η 2 s ) (The words in brack ets sho w the corresp onding gluings.) T o find endomorphisms of these ato ms, note that there a r e triangles S 3 ∨ S 4 (2 r η ) − − − → S 3 → C 5 (2 r η ) → S 4 ∨ S 5 , (3.5) S 4 0 @ η 2 s 1 A − − − → S 3 ∨ S 4 → C 5 ( η 2 s ) → S 5 , (3.6) S 3 ∨ S 4 0 @ 2 r η 0 2 s 1 A − − − − − − − →→ C 5 (2 r η 2 s ) → S 3 ∨ S 4 . (3.7) By Theorem 1.1, Es ( C 5 (2 r η )), up to a n ideal I such that I 2 = 0 , is isomorphic to the endomorphism r ing of the map f = (2 r η ) in the category Bim ( S ) / J . An endomorphism of f in Bim ( S ) is a pair ( α, β ), where α =  a bη 0 c  ( a, c ∈ Z , b ∈ Z / 2 ) , β ∈ Z , suc h that β f = f α , i.e. β = a ≡ c mo d 2. Moreov er, one easily sees that J consists o f the pairs with the first comp onen t  2 r x xη 0 0  , whence Es ( C 5 (2 r η )) /I 2 is isomorphic to the subring of Z / 2 r +1 ⊕ Z consist- ing of all pairs ( a, c ) with a ≡ c mo d 2 . This ring has no non triv- ial idemp oten t, hence, C 5 (2 r η ) is indeed indecomp osable, hence, an atom. Moreov er, using the triangle (3.5), one can see that I ≃ Z / 2 and 14 YU RIY A. DROZD Es ( C 5 (2 r η )) is isomorphic to the ring of triangular matrices  a b 0 c  , where a ∈ Z / 2 r +1 , b ∈ Z / 2 , c ∈ Z , a ≡ c mo d 2. The same re- sult for Es ( C 5 ( η 2 r )) follow s from the triangle (3.6). Fina lly , one gets from the tria ngle (3.7) that Es ( C 5 (2 r η 2 s )) is isomorphic to the ring of triangular matrices  a b 0 c  , where a ∈ Z / 2 r , b ∈ Z / 2 , c ∈ Z / 2 s , a ≡ c mo d 2. Therefore these p o lyhedra are also atoms. They are called Chang atoms Moreo v er, the last ring is lo cal, t h us the m ulti- plicit y of C 5 (2 r η 2 s ) (as we ll as o f an y its shift) in a decomp osition of a p olyhedron in t o a b ouquet of indecompo sables is t he same for all suc h decomp ositions. Not e tha t the same is true for sus p ended atoms M d ( q ). On the other ha nd, the triangles (3.5) and (3 .6) imply that H 3 ( C 5 (2 r η ) ≃ H 4 ( C 5 ( η 2 r ) ≃ Z / 2 r , while o ther homologies of these spaces are zero. Altogether, it giv es the f ollo wing description of the category S 3 . Theorem 3.4 (Whitehead–Chang, [25, 11]) . Any p olyhe dr on fr om S 3 uniquely ( up to p ermutation of summands) de c omp oses into a b ouquet of spher es S 3 , S 4 , S 5 , susp ende d Mo or e atoms M 4 ( q ) , M 5 ( q ) and Chang atoms C 5 ( η ) , C 5 (2 r η ) , C 5 ( η 2 s ) and C 5 (2 r η 2 s ) . Using terms fr o m the represen tation theory , one can sa y that the categories S n , n ≤ 3, are discr ete (or essential ly finite ). In this con- text it means that there are only finitely many isomorphism classes of p olyhedra in S n with a prescribed expo nen t of the torsion part of homologies. (So it lo oks similar to the description of finitely generated ab elian groups.) 4. T ame case: Ba ue s–Hennes Theorem W e study no w the category S 4 . By Theorem 2.2, S 4 = A † B , where A = S 3 S 2 , B = S 2 S 2 . By Theorem 3.1, ev ery p olyhedron from A (from B ) is a b ouquet of spheres S 5 , S 6 and Mo o re atoms M 6 ( q ) (resp ectiv ely , S 4 , S 5 and M 5 ( q )). W e ha v e already calculated morphisms b etw een indecomposables in S 2 . Just in the same w a y one calculates morphisms fr o m the ob jects of S 3 S 2 and those of S 2 S 2 . W e omit the details, which are standard; the result is presen ted in T able 1. Actually , the groups Hos ( M 6 s , M 5 r ) can b e naturally c o nsidered as the groups of upp er triangular matrices  a b 0 c  o ve r Z / 2 with b = 0 if s = r = 1 . Again the sum of morphisms corresp ond to the usual sum of matrices, with the exceptions for s > 1 , r = 1 and s = 1 , r > 1, when MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 15 T able 1. S 5 S 6 M 6 1 M 6 s ( s > 1) S 4 Z / 2 Z / 2 Z / 4 Z / 2 ⊕ Z / 2 S 5 Z Z / 2 Z / 2 Z / 2 M 5 1 Z / 2 Z / 4 Z / 2 ⊕ Z / 2 Z / 4 ⊕ Z / 2 M 5 r ( r > 1) Z / 2 Z / 2 ⊕ Z / 2 Z / 2 ⊕ Z / 4 Z / 2 ⊕ Z / 2 ⊕ Z / 2 the sum of matrices m ust b e t wisted as fo llo ws:  a b 0 c  +  a ′ b ′ 0 c ′  =  a + a ′ b + b ′ + aa ′ 0 c + c ′  if s > 1 , r = 1 ,  a b 0 c  +  a ′ b ′ 0 c ′  =  a + a ′ b + b ′ + cc ′′ 0 c + c ′  if s = 1 , r > 1 . The m ultiplication of elemen t s f rom Hos ( M 6 s , M 5 r ) b y morphisms b e- t w een ob jects from A and B (also presen t ed b y triangula r matr ices as in Section 3) corresp ond to the usual pro duct of matrices. Therefore, a morphism A → B can b e naturally considered as a blo c k matrix presen ted in T able 2. In this ta ble a sym b ol 2 ( ∞ ) sho ws that the T able 2. x                                                   ( 1 ) ( 2 ) ( 3 ) . . . . . . ( 3 ) ( 2 ) ( 1 ) ( 1 ) 2 2 2 . . . 2 2 . . . 2 2 0 ( 2 ) 2 2 2 . . . 2 2 . . . 2 2 2 ( 3 ) 2 2 2 . . . 2 2 . . . 2 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 . . . 2 2 . . . 2 2 2 0 0 0 . . . ∞ 2 . . . 2 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 ) 0 0 0 . . . 0 2 . . . 2 2 2 ( 2 ) 0 0 0 . . . 0 2 . . . 2 2 2 ( 1 ) 0 0 0 . . . 0 2 . . . 2 2 2                       − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − → corresp onding blo c k has v alues from Z / 2 (resp ectiv ely , from Z ). Zeros sho w that the corres p onding blo c k is alw ay s zero. Arrows on the left 16 YU RIY A. DROZD and b elo w sym b olize the action o f morphisms b et w een the ob jects from A and B resp ectiv ely . The lab els ( 1 ) , ( 2 ) , . . . (or ( 1 ) , ( 2 ) , . . . ) sho w that the corresp onding horizon ta l (resp ectiv ely , v ertical) strip es are of the same size and we must use the same elemen tary transformations in b oth o f them. These strip es corresp ond to M d r with the same d and r . Note that there are 2 horizon tal and 2 v ertical strip es without suc h lab els. They corresp ond to sphere s S d . This matrix problem is a sligh t v ariatio n of a w ell-kno wn one, namely , represen tations of bunches of chains (see [9] or [10, App endix B]) . It im- plies a description of indecomposable ob jects in the category Bim ( S 4 , 2 ), hence, in S 4 . W e call them strings and b ands , as it is usual in the rep- resen tation theory of algebras. No t providing details (see [15]), w e just presen t the corresp o nding att a c hmen t diagrams (T able 3). It is conv e- nien t to distinguish tw o ty p es of strings: usual and de c or ate d ; I hope that the pictures show the difference. “Decorations” (one for eac h string) are sho wn with do uble lines. W e o mit integers precising the degrees of “vertical” attac hmen ts, as w ell as one precis ing the “long” attac hment in a decorated strings of the first kind; they can b e arbi- trary and differ f o r differen t attachmen ts. Certainly , each diagram is actually finite: it starts at a ny place on the left and stops at an y place on t he right. Multiple bullets in the case of bands sym b olize not a unique cell but sev eral (sa y m ) copies of it (the same for each ball). All atta chmen ts except the one marked b y the w a vy line are “natural”: the first copy of an upp er cell is attached t o the first copy o f a lo w er one, the second to the second, etc. The a t t ac hmen t mark ed by the w avy line is “twis ted” b y an in vertible F rob enius matrix Φ of size m × m ov er the field Z / 2 with the c haracteristic p olynomial f ( x ), whic h must b e a p ow er of an ir reducible one and suc h that f (0) 6 = 0. F or instance, if f ( x ) = x 3 + x + 1, i.e. m = 3 and Φ =   0 0 1 1 0 0 0 1 1   , this att a c hmen t is: 6 • $ $ H H H H H H H H H H H H H H • $ $ H H H H H H H H H H H H H H • u u j j j j j j j j j j j j j j j j j j j j j j j j   4 • • • One can che ck t ha t all strings and ba nds are indecomp osable and pairwise non-isomorphic. Note also that all atoms from S 4 are p - primary (2- primary , except Mo ore atoms M d ( p r ) with o dd p , whic h are p -primary). Therefore, we hav e the uniqueness of decomp osition of spaces from S 4 in to b ouquets of susp ended atoms. So w e get the follo wing result. W e call strings and bands B aues atoms . MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 17 T able 3. usual strings 7 • • • 6 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • C C C C C C C C C C · · · 5 •                 •                 •                 · · · 4 • C C C C C C C C C C • • deco rated strings 7 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • • 6 • { { { { { { { { { { •                 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • C C C C C C C C C C 5 • • •                 · · · · · · 4 • •                                           • and 7 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 •                                 •                 · · · 6 •                 •                                 • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • C C C C C C C C C C 5 • C C C C C C C C C • • • · · · 4 • • • bands 7 • • • • • • • • • 6 • • • 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • • • I I I I I I • • • · · · 5 • • •                 • • •                 • • •                 4 • • • Φ 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s 3s • • • • • • J J J J J J J J J J J Theorem 4.1 (Baues–Hennes [8]) . Any p ol yhe dr on fr om S 4 de c om- p os es uniquely into a b ouquet of spher es, susp ende d Mo o r e atoms , sus- p en de d Chang atoms and Baues atoms. 18 YU RIY A. DROZD In Section 7 w e shall see that actually S 4 is t he last case where a “g o o d” description of p o lyhedra is p ossible. Starting from S 5 this problem becomes wild . 5. Torsion free pol yhe dra. Finite case Consider now torsion free case. Note that if all H k ( X ) are torsion free, the atta chmen t dia gram cannot contain “Mo o re frag men ts” • • In particular, among the at oms from Sections 3 and 4 only Chang atom C 5 = C 5 ( η ) and the double Chang atom C 7 2 = C 7 ( η 2 ) with the attac hment diag ram 7 • 6 5 4 • are torsion free. Therefore, if w e set in Theorem 2.4 n = 5 , m = 4, the category A 0 consists of b ouquets of spheres S 8 and the cate- gory B 0 consists of b ouquets of spheres S d (5 ≤ d ≤ 8), susp ended Chang atoms C 7 , C 8 and susp ended double Chang atoms C 8 2 . Ob vi- ously , S 0 ( S 8 , S 8 ) = 0. Easy calculation giv e the f o llo wing v alues o f the groups Γ( B ) = S 0 ( S 8 , B ) for ato ms B from B 0 : B S 5 S 6 S 7 C 7 C 8 C 8 2 Γ Z / 24 Z / 2 Z / 2 Z / 12 0 Z / 1 2 Morphisms of these spaces induce monomorphisms Γ( S 7 ) → Γ( C 7 ) → Γ( S 5 ) and Γ( S 6 ) → Γ( S 5 ), epimorphism Γ( S 5 ) → Γ( C 8 2 ), and isomor- phisms Γ( S 7 ) → Γ( S 6 ) and Γ( S 7 ) → Γ( S 6 ). Th us, an o b ject f r om Bim ( S 0 5 , 4 ) can be pre sen ted b y a blo c k matrix as in T able 4. Here in- side eac h blo c ks w e ha ve written the groups, wherefrom the co efficien ts of this blo ck are. The arrows show the allo wed tra nsformation b etw een blo c ks. An in teger k in the arrow s p oin t out tha t, when w e p erform this transformation, the ro w mus t b e mu lt iplied by k . (No integer means that k = 1 .) F or instance, we can a dd the row s of the third ma t r ix m ultiplied b y 2 to the ro ws of the first one. Certainly , comp ositions of these transformations a re also allo wed. Th us, for instance, w e can add the ro ws of t he third matrix m ultiplied b y 2 to the rows of the second one to o. The ar ising matrix problem is rather simple. It is of finite t yp e, and T able 6 shows the attac hment diagrams of the corresp onding atoms from T 5 . W e call them A-atoms of the 1st kind . The inte g er v sho w, whic h m ultiple of the generator o f the group π S 8 ( S 5 ) ≃ Z / 2 4 is MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 19 T able 4.   Z / 24   Z / 12 Z / 12 2 O O   Z / 2 6 O O Z / 2 12 R R T able 5. 9 • •                  •                   •                  •                   8 • 7 • • 6 • 5 • • • • • A ( v ) A ( η v ) A ( η 2 v ) A ( v η ) A ( v η 2 ) 9 •                   •                   •                   •                   8 • • 7 • • • • 6 • • 5 • • • • A ( η 2 v η ) A ( η 2 v η 2 ) A ( η v η ) A ( η v η 2 ) used for the “ long” attachme nt. Actually , 1 ≤ v ≤ 12 in the case of A ( v ), 1 ≤ v ≤ 3 in the case of A ( η v η ), 1 ≤ v ≤ 6 in all other cases. So we hav e got a description of polyhedra from T 5 . Theorem 5.1 (Baues–Drozd [4]) . Every p olyhe dr on fr om T 5 is a b ou- quet of spher es, susp ende d Chan g and double Ch a ng atoms, a n d the A-atoms of the first kind. 20 YU RIY A. DROZD Note that this time the decomp osition is not unique; ev en the can- cellation law do es not hold. F or instance, A (3) ⊕ S 5 ≃ A (9) ⊕ S 5 [4, 1 5 ]; see ibidem more on decomposition law s. Analogous is the case o f T 6 , when w e tak e m = 4. W e omit details, just sc hematically presen ting the arising matrix problem in T a ble 6. The da shed line from the 4th to the 6th lev el show the transforma- tion that only acts on the left-hand column (on Z / 2 comp onen ts). The resulting list of atoms (their a ttac hmen t diagrams) see in the T able 6.   Z / 24 0   Z / 12 0 Z / 12 0 2 O O         $ ' * Z / 2 Z / 2 6 O O _ 6   12   0 Z / 2   Z / 2 Z / 24 12 O O 0 Z / 12 2 O O T able 7. W e call them A-atoms of the se c ond kind . The inte g ers v and w sho w, as ab ov e, the m ultiple o f generator, resp ective ly , of π S 9 ( S 6 ) and π S 10 ( S 7 ) used for the cor r esp onding a ttac hmen ts. In all cases v , w ∈ { 1 , 2 , 3 , 4 , 5 , 6 } . So we hav e got a description of polyhedra from T 6 . Theorem 5.2 (Baues–Drozd [7]) . Every p olyhe dr on fr om T 6 is a b ou- quet of spher es, susp ende d Chang and double Ch a ng atoms, susp ende d A-atoms of the first kind and A-atoms of the se c ond kind . In the next section w e shall use the v alues o f Hos -groups b et wee n Chang at o ms a nd spheres. T o deal with the Chang atom C 5 w e apply the bifunctor Hos to the cofibratio n seq uence (5.1) S 4 η − → S 3 → C 5 → S 5 η − → S 4 . MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 21 T able 7. 11 •                  •                  •                  •                  •                  10 •                  •                  •                  •                  •                  9 • • • 8 • • • 7 • • • • • 6 • • • • • A ( v η 2 w ) A ( η v η 2 w η ) A ( η 2 v η 2 w η 2 ) A ( v η 2 w η ) A ( η v η 2 w ) 11 •                  •                  •                  •                  10 •                  •                  •                  •                  9 • • • 8 • • • 7 • • • • 6 • • • • A ( v η 2 w η 2 ) A ( η 2 v η 2 w ) A ( η 2 v η 2 w η ) A ( η v η 2 w η 2 ) It g ives the comm utativ e dia g ram with exact ro ws and columns (w e write here ( X , Y ) instead of Hos ( X , Y ) ) Z − − − → Z / 2 − − − → ( C 5 , S 4 ) − − − → 0 − − − → Z   y   y   y   y   y Z / 2 − − − → Z / 2 − − − → ( C 5 , S 4 ) − − − → Z − − − → Z / 2   y   y   y   y   y ( S 4 , C 5 ) − − − → ( S 5 , C 5 ) − − − → ( C 5 , C 5 ) − − − → ( S 3 , C 5 ) − − − → ( S 4 , C 5 )   y   y   y   y   y 0 − − − → Z − − − → ( C 5 , S 5 ) − − − → 0 − − − → 0   y   y   y   y   y Z − − − → Z / 2 − − − → ( C 5 , S 4 ) − − − → 0 − − − → Z , 22 YU RIY A. DROZD where all maps Z → Z / 2 are surjec tive and all maps Z / 2 → Z / 2 are bijectiv e. It giv es the follo wing v alues of Hos -groups: Hos ( C 5 , S 4 ) = Hos ( S 4 , C 5 ) = 0 Hos ( S 3 , C 5 ) = Hos ( C 5 , S 5 ) = Z , Hos ( C 5 , S 3 ) = Hos ( S 5 , C 5 ) = 2 Z , Hos ( C 5 , C 5 ) = D , where D (the “ dyad ”) is the subrings of Z × Z consisting of all pairs ( a, b ) with a ≡ b mo d 2. Similar observ a tions applied to the susp ended v ersions of the se- quence (5.1) and the cofibration seq uence S 6 η 2 − → S 4 → C 7 2 → S 7 η 2 − → S 5 giv e T able 8 of the v alues Hos ( X , Y ) for susp ended atoms from T 4 . In T able 8. S 4 C 7 2 : 4 7 C 6 : 4 6 S 5 C 7 : 5 7 S 6 S 7 S 4 Z 2 Z Z / 12 2 Z 0 Z / 2 0 Z / 12 Z / 2 Z / 24 C 7 2 : 4 Z Z = Z / 12 2 Z 0 Z / 2 0 Z / 12 0 Z / 12 7 0 0 Z = 0 0 0 0 Z 0 2 Z C 6 : 4 Z Z Z / 12 Z = 0 0 0 Z / 12 0 2 Z 6 0 0 0 0 Z = 0 0 0 2 Z 0 S 5 0 0 0 0 0 Z 2 Z 0 Z / 2 Z / 2 C 7 : 5 0 0 0 0 0 Z Z = 0 0 0 7 0 0 2 Z 0 0 0 0 Z = 0 2 Z S 6 0 0 Z / 2 0 Z 0 0 0 Z Z / 2 S 7 0 0 Z 0 0 0 0 Z 0 Z this table the Hos -groups f o r susp ended Chang atoms a r e presen ted in matrix fo rm, emphasizing whic h comp onen ts ha v e come from the ce lls of giv en dimensions. The sup erscripts = sho w that the diagonal parts of the corresp onding matrices are with en tries not f rom Z × Z , but from MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 23 D . F or instance, Es ( C 7 2 ) is presen ted as the r ing of triangular matrices  a b 0 c  , w here a, c ∈ Z , a ≡ c mo d 2 , b ∈ Z / 12. Under suc h prese n- tation the m ultiplication o f morphisms turns in to the multiplication of matrices. 6. Torsion free pol yhe dra. T ame case The category T 7 is more complicated. T o describ e it, we us e The- orem 2.4 with n = 7 , m = 3. Then A 0 consists of the b ouquets of spheres S d (10 ≤ d ≤ 12) and susp ended Chang atoms C 12 , while B 0 consists of b o uquets of spheres S d (7 ≤ d ≤ 1 0 ) susp ended Chang atoms C 9 , C 10 and susp ended double Chang atoms C 10 2 . The calcu- lations similar to those of the end of preceding section giv e T a ble 9 of the v alues of gro ups S 0 ( X , Y ) f or the susp ended a toms X ∈ A 0 and Y ∈ B 0 , also presen ted in matrix fo rm. The superscripts ∗ sho w T able 9. S 10 S 11 S 12 C 12 : 10 12 S 7 Z / 24 0 0 Z / 24 0 C 10 2 : 7 0 0 Z / 2 Z / 24 ∗ 0 10 0 0 Z / 2 0 Z / 2 ∗ C 9 : 7 Z / 12 0 0 Z / 24 ∗ 0 9 0 0 Z / 24 0 Z / 2 ∗ S 8 Z / 2 Z / 24 0 0 0 C 10 : 8 0 Z / 12 0 0 0 10 0 0 0 0 0 S 9 Z / 2 Z / 2 Z / 24 0 Z / 12 S 10 0 Z / 2 Z / 2 0 0 that in the corresp onding groups w e iden tify the elemen ts of order 2. So actually , these v a lues are isomorphic to Z / 24, but it is c on ve nient to consider them as ( Z / 24 ⊕ Z / 2) / (12 , 1). Then again the action of 24 YU RIY A. DROZD morphisms from A 0 and B 0 , as presen ted in T able 8 (or, rat her, its suspended vers io n) turns in to the m ultiplication of matrices. Again w e obtain a bimo dule problem close to that of bunc hes of chains , esp e- cially , in its “ de c or ate d ” v ersion (see [17]). T o pres ent the answ er (f or details se e [16]), w e intro duce the following notations and definitions. Definition 6.1. (1) W e consider c ha ins E k and F k (1 ≤ k ≤ ): E 1 = { e 1 < e 2 < e 4 } , F 1 = { f 4 < f 1 } , E 2 = { e 5 < e 9 } , F 2 = { f 3 < f 5 } , E 3 = { e 6 < e 7 } , F 3 = { f 2 } , E 4 = { e 3 < e 10 < e ′ 9 < e ′ 6 } , F 4 = { f ′ 1 < f ′ 2 < f ′ 3 } . Actually , the elemen ts e i ( f j ) corresp ond to the rows (columns) of T able 9, while the relations c − correspond to the elemen ts of the groups C 0 ( A, B ). W e need extra elemen ts e ′ i and f ′ j since the entries Z / 2 in this table b ehav e in a differen t wa y than the other ones. W e set E = S 4 i =1 E i , F = S 4 i =1 F i , X = E ∪ F . x ≈ y means that x a nd y b elong to t he same set E i or F i . (2) W e define symmetric relations ∼ and − o n X setting x − y if x ∈ E i , y ∈ F i or vice v ersa; e i ∼ e ′ i ( i ∈ { 6 , 9 } , f j ∼ f ′ j (1 ≤ j ≤ 3 ). W e also define the symmetric relatio ns c − , where c ∈ { 1 , 2 , 3 , 4 , 6 } , setting e i c − f j if e i − f j and the ( ij )- th entry in T able 9 is Z /m with c | m . W e denote b y R the set of all relations {∼ , c −} and by v ( c ) t he biggest d suc h that 2 d divides c . (3) W e define a wor d as a sequence w = x 1 r 2 x 2 r 3 . . . r l x l where x i ∈ X , r i ∈ R suc h that (a) x k − 1 r k x k in X for eac h 1 < k ≤ l ; (b) if r k = ∼ , then r k +1 = c − and vice v ersa; (c) if r 2 = c − (respectiv ely , r l = c − ), there is no elemen t y ∈ X suc h that x 1 ∼ y (respectiv ely , x l ∼ y ); (d) if r k = c − with v ( c ) = 1 , then either 2 < k < l , or k = 2 , x 1 = e 1 , or k = l , x l = x 1 ; MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 25 (e) if r = c − with v ( c ) = 2, then r can only o ccur in the follow - ing w ords or their rev erse: e 4 ∼ e 5 r f 3 ∼ . . . (of an y length) , e 1 r f 4 ∼ f 5 , e 3 ∼ e 2 r f 4 ∼ f 5 , . . . c ′ − e 4 ∼ e 5 r f 3 ∼ . . . (of an y length) , e 1 r f 4 ∼ f 5 c ′ − . . . (of an y length) , e 3 ∼ e 2 r f 4 ∼ f 5 c ′ − . . . (of an y length) , e 1 r f 4 ∼ f 5 c ′ − . . . (of an y length) , e 1 r f 1 ∼ f ′ 1 , e ′ 6 ∼ e 6 r f 2 ∼ f ′ 2 , e ′ 9 ∼ e 9 r f 3 ∼ f ′ 3 , where c ′ ≡ 0 (mo d 3); (f ) if w con ta ins a subw ord e i c − f j , c ∈ { 3 , 9 } or its re- v erse, it do es not con tain any sub w ord e i ′ c ′ − f j ′ , c ′ 6≡ 0 (mo d 3) , e i ≈ e i ′ (equiv alen tly , f j ≈ f j ′ ) o r its rev erse. Here the r everse to the w ord w is the w o rd w ∗ = x l r l x l − 1 . . . x 2 r 2 x 1 . W e call l the length of the w ord w . (4) W e define a cycle as a pair z = ( w , r 1 ), where w is a w ord suc h that r 2 = r l = ∼ and r k 6 = c − with v ( c ) = 2, while r 1 = c − with v ( c ) 6 = 2 and x l r 1 x 1 in X . F or suc h a cycle w e set x q l + k = x k and r q l + k = r k for an y q and 1 ≤ k ≤ l . (5) The m -th shift of the cycle z = ( w , r 1 ) is defined as the cycle z ( m ) = ( w ( m ) , r 2 m +1 ), where w ( m ) = x 2 m +1 r 2 m +2 x 2 m +2 . . . r 2 m x 2 m . (6) A cyc le ( w , r 1 ) is called p erio dic if w is of the for m w = v r 1 v r 1 . . . r 1 v for a shorter cycle ( v , r 1 ). (7) W e call t w o w ords, w a nd w ′ = x 1 r ′ 2 x 2 r ′ 3 . . . r ′ l x l (with the same x k ), e lementary c ongruent if there are t wo indices k 1 , k 2 suc h that r k 1 = 3 c − , r k 2 = d − for some c 6 = 3 , d 6 = 3 , r ′ k 1 = c − , r ′ k 2 = 3 d − , r ′ k = r k for k / ∈ { k 1 , k 2 } , x k 1 ≈ x k 2 or x k 1 ≈ x k 2 − 1 . (8) W e call t wo w ords w , w ′ c on gruent and write w ≡ w ′ if there is a sequence of w ords w = w 1 , w 2 , . . . , w n = w such that w k and w k +1 are elemen ta r y congruen t for 1 ≤ k < n . W e call t wo cycles z = ( w , r 1 ) and z ′ = ( w ′ , r ′ 1 ) c ongruent and write z ≡ z ′ if w ′ ≡ z and r ′ 1 = r 1 . 26 YU RIY A. DROZD W e recall that t w o p o lyhedra X, Y are called c ongruent if X ∨ Z ≃ Y ∨ Z for some polyhedron Z . Then w e write X ≡ Y . Theorem 6.2. (1) Every wor d w define s an inde c omp osable p oly- he d r on P ( w ) fr om T 7 , c al le d string po lyhedron . (2) L et π ( t ) 6 = t b e an irr e ducible p olynomia l over the field Z / 2 . Every triple ( z , π ( t ) , m ) , wher e is a non-p eri o dic cycle and m ∈ N , define s an inde c omp osable p olyhe dr on P ( z , π , m ) fr om T 7 , c al le d band p olyhedron . (3) Every inde c omp osable p olyhe dr on fr om T 7 is c ongruent either to a string or to a b and o ne. (4) P ( w ) ≡ P ( w ′ ) if and only if either w ′ ≡ w or w ′ ≡ w ∗ . (5) P ( z , π ( t ) , m ) ≡ P ( z ′ , π ′ ( t ) , m ) if an d o nly i f m = m ′ and one of the fol low i n g p ossibilities hold: (a) π ′ ( t ) = π ( t ) and either z ′ ≡ z ( k ) with k even or z ′ = z ∗ ( k ) with k o dd; (b) π ′ ( t ) = t d π (1 /t ) , wh er e d = deg π , and either z ′ = z ( k ) with k o dd or z ′ = z ∗ ( k ) with k ev en. (6) Neither string p ol yhe dr on is c ongruent to a b and on e. (7) The cofibrat ion seq uence A f − → B → C f → S A, A ∈ A 0 , B ∈ B 0 , and the att a c hmen t diag ram of a string p olyhedron P ( w ) is constructed as follo ws. (1) The indecomp osable s ummands of A corresp ond to the follow - ing sub w ords of w (o r their rev erse): S 10 to f 1 ∼ f ′ 1 , S 11 to f 2 ∼ f ′ 2 , S 12 to f 3 ∼ f ′ 3 , C 12 to f 4 ∼ f 5 . (2) The indecomp osable summands of B corr esp ond to the follo w- ing sub w ords of w (o r their rev erse): S 7 to e 1 , C 10 2 to e 2 ∼ e 3 , C 9 to e 4 ∼ e 5 , S 8 to e 6 ∼ e ′ 6 , C 10 to e 7 , S 9 to e 9 ∼ e ′ 9 , S 10 to e 10 . MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 27 (3) The attac hmen t s corresp ond to the sub words e i c − f j (or their rev erse). Namely , such a n attac hment starts at the f -end of the corresp onding sub w or d and ends at it s e -end; t he n um b er c sho ws whic h m ultiple of the generator of the ( ij ) - th group from T able 9 m ust b e tak en. F or instance , if w = e 10 1 − f ′ 2 ∼ f 2 8 − e 6 ∼ e ′ 6 1 − f ′ 1 ∼ f 1 2 − e 4 ∼ e 5 6 − f 5 ∼ f 4 1 − e 2 ∼ e 3 1 − f ′ 3 ∼ f 3 2 − e 5 ∼ e 4 3 − f 1 ∼ f ′ 1 1 − e ′ 9 ∼ e 9 12 − f 3 ∼ f ′ 3 , the polyhedron P ( w ) has the attac hmen t diagram 13 • 6                 • 2               • 12                 12 • 8 * * * * * * * * * * * * * * * * 11 • 2 * * * * * * * * * * * * * * * * • 1 * * * * * * * * * * * * * * * * • 3                 10 • • 9 • • • 8 • 7 • • • Let no w P ( z , π ( t ) , m ) b e a band p olyhedron. Replacing w b y w ∗ , w e ma y supp ose that x 1 ∈ E , x n ∈ F . Let also Φ b e the F rob enius matrix with the c har a cteristic po lynomial π ( t ) m . Then the cofibration sequence and the attach men t diagram are constructed as follows . (1) Do the construction as ab o ve for the word w . (2) Replace ev ery summand A j of A and ev ery summand B i of B b y m copies of it, A j 1 , . . . , A j m and B i 1 , . . . , B im . (3) If there w as an attac hmen t A j c − → B i , replace it b y the attac h- men ts A j k c − → B ik (1 ≤ k ≤ m ). (4) If A j is the last summand of A , B i is the first summand of B and r 1 = c − , add new attac hmen t s A j k c − → B il in all cases, when the ( lk )- th co efficien t of the matrix Φ is nonzero. F or instance, consider the band p olyhedron P ( z , t 2 + t + 1 , 3) z = ( w , 1 − ), where w = e 2 ∼ e 3 1 − f ′ 3 ∼ f 3 2 − e 9 ∼ e ′ 9 1 − f ′ 1 ∼ f 1 3 − e 4 ∼ e 5 6 − f 5 ∼ f 4 . 28 YU RIY A. DROZD Then the at t a c hmen t diagram is 13 • • •• 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • • •• 6                                     12 11 • • •• 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • • •• 10 • • •• 9 • • •• • • •• 8 7 • • •• 1 -m -m .n .n /o /o 0p 0p 1q 1q 2r 3s 3s 4t 5u 8x 8x 9y :z ;{ <| =} >~ ? @ A B C D D • • •• Here the double lines sho w the attac hmen ts lik e • • • • • • • • while the wa vy line show s the attachm ent • J J J J J J • J J J J J J • J J J J J J • t t t t t t f f f f f f f f f f f f f f f • • • • ruled b y the F r ob enius matrix with the c hara cteristic p olynomial π ( t ) 2 = t 4 + t 2 + 1, namely ,     0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0     . 7. Wild cases Since w e a re dealing with additive categories that are not categories o ve r a filed, w e hav e to precise the notion of wildness. The follo wing one see ms to w ork in all kno wn cases. Definition 7.1. W e call an additiv e categor y C wild if, there is a field k suc h that for ev ery finitely generated k -algebra Λ there is a full sub category C Λ ⊆ S and an epiv alence C Λ → Λ- mo d (the category of Λ-mo dules that a r e finite dimensional o ver k ). One can see that for alg ebras ov er a field this definition is equiv alen t to the usual one ( see, for instance [14]). One can also easily sho w that if a catego ry D is wild and there is an epiv alence C ′ → D for a full sub category C ′ ⊆ C , then C is wild as we ll. No w w e presen t the results on wildness of categories S n and T n . Theorem 7.2 (Baues [6]) . If n > 4 , the c ate go ry S n is wild. Pr o of. Ob viously , one o nly has to prov e the claim f or n = 5. The category S 5 con tains the full sub category C = A † B , where A consists of b ouquets of suspended Mo ore atom A = M 6 (2) and B consists of b ouquets of susp ended Mo ore atoms B = M 8 (2).Let V = A S B . Since MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 29 Hos ( B , A ) = 0, Corollar y 1.2 is applicable. Moreov er, t he ideal J in this case is zero, so C / I ≃ Bim ( V ) with I 2 = 0, henc e, the natura l functor C → Bi m ( V ) is an epiv alence. Consider the cofibratio n sequence (7.1) S 7 2 − → S 7 → A → S 8 2 − → S 8 . Apply to it the functors Hos ( , S 6 ) and Hos ( , S 5 ). T aking in to accoun t the Hopf map η : S 6 → S 5 w e get the comm utative diagram with exact ro ws 0 − − − → Z / 2 − − − → Hos ( A, S 6 ) − − − → Z / 2 − − − → 0 η ∗   y   y   y ≀ 0 − − − → Z / 2 − − − → Hos ( A, S 5 ) − − − → Z / 2 − − − → 0 . Since η 3 = 4 ν , where ν is the elemen t of order 8 in Hos ( S 8 , S 5 ), the map η ∗ in this diagram is zero, therefore, t he lo wer exact sequence splits and Hos ( A, S 5 ) ≃ Z / 2 ⊕ Z / 2.Quite similarly , one sho ws that Hos ( S 8 , B ) ≃ Z / 2 ⊕ Z / 2. No w apply t he functors Hos ( , S 5 ) and Hos ( , B ) to t he exact sequence (7.1) and take in to accoun t t he map S 5 → B form the definition of B = M 6 (2). Since Hos ( S 7 , B ) ≃ Z / 2, w e get the comm utativ e diagram with exact ro ws 0 − − − → Z / 2 − − − → Hos ( A, S 5 ) − − − → Z / 2 − − − → 0   y   y   y ≀ 0 − − − → Z / 2 ⊕ Z / 2 − − − → Hos ( A, B ) − − − → Z / 2 − − − → 0 . W e kno w that the upp er ro w of this dia g ram splits. Hence, the lo w er ro w splits to o, so Hos ( A, B ) ≃ ( Z / 2) 3 . Recall that Es ( A ) ≃ Es ( B ) ≃ Z / 4 (Prop osition 3.3). Hence, there is an epiv alence Bi m ( V ) → Λ- mo d , where Λ is the path algebra of the quiv er • ( ( / / 6 6 • o ve r the field Z / 2. The la t ter is w ell- known to b e wild, therefore, so is also S 5 .  Theorem 7.3 ([16]) . The c ate g ory T n is wild for n > 7 . Pr o of. Again w e only ha ve to prov e it fo r n = 8. The category T 8 con- tains the full sub category C = A † V B , w here A consis ts o f b o uquets of Chang at oms C 1 4 2 and B consists of b ouquets of spheres § 8 and S 1 1, and V = B S 0 8 , 3 A . Moreo ve r, I 0 8 , 3 ∩ Bim V = 0, so t here is an epiv alence C → Bim ( V ). Consider the cofibratio n seq uence S 13 η 2 − → S 11 → C 14 2 → S 14 → S 12 and apply to it the f unctor Hos ( , S 11 ). W e get the exact sequence Z / 2 ( η 2 ) ∗ − − − → Z / 24 → Hos ( C 14 2 , S 11 ) → Z → Z / 2 , 30 YU RIY A. DROZD wherefrom S 0 ( C 14 2 , S 1 1) ≃ Z / 12. Moreov er, there is a comm ut a tiv e diagram of cofibration sequences S 13 η − − − → S 12 − − − → C 14 − − − → S 14 η − − − → S 13 Id nol imits   y η   y   y   y Id nolimits   y η S 13 η 2 − − − → S 11 − − − → C 14 2 − − − → S 14 − − − → η 2 S 12 . Applying t he f unctor Hos ( , S 8 ), w e get the commutativ e diagram with exact ro ws 0 − − − → Z / 2 − − − → Hos ( C 14 2 , S 8 ) − − − → Z / 24 − − − → 0   y   y   y 0 − − − → Z / 2 − − − → Hos ( C 14 , S 8 ) − − − → 0 − − − → 0 . (Recall that π S d +4 ( S d ) = π S d +5 ( S d ) = 0 and π S d +6 ( S d ) = Z / 2 [24]). Therefore, S 0 ( C 14 2 , S 8 ) ≃ Z / 24 ⊕ Z / 2. So w e presen t maps a ∈ V ( A, B ), where A ∈ A , B ∈ B , as blo c k-triangular matrices a =  a 1 a 2 0 a 3  , where a 1 is with the co efficien ts from Z / 24 , a 2 is with co efficien ts from Z / 2 and a 3 with co efficien ts f r o m Z / 12. On the other ha nd, maps α : A → A ′ , where A, A ′ ∈ A , and β : B → B ′ , where B , B ′ ∈ B can b e presen ted b y block- triangula r matrices α =  α 1 α 2 0 α 3  and β =  β 1 β 2 0 β 3  , where α 2 has co efficien ts f rom Z / 12, β 2 has co efficien ts f rom Z / 24, other blo c ks ha ve comp onen ts from Z and α 1 ≡ α 3 mo d 2. W e consider the full subcatego ry C ⊂ Bim ( V ) consisting o f all ma ps a suc h tha t the corresponding blo c ks a 1 , a 2 , a 3 are of the form a 1 =  6 I 0 0 0 12 0  , a 2 =  0 I 0 0 0 u  a 3 =  6 v 1 6 v 2 0  , where the entries I stand for identit y matrices (not necessary of the same dimensions) and u, v 1 , v 2 are arbitra r y matrices with coefficien ts from Z / 2 of pro p er sizes. W e write a = a ( u, v 1 , v 2 ). One can v erify that if ( α , β ) is a morphism a ( u, v 1 , v 2 ) → a ( u ′ , v ′ 1 , v ′ 2 ), there are inte- gral matrices γ 1 , γ 2 , γ 3 suc h that v i γ 1 = γ 2 v i ( i = 1 , 2) and uγ 3 = γ 1 u . Con v ersely , any giv en triple γ 1 , γ 2 , γ 3 with these pro p erties can b e ac- complished to a morphism a ( u, v 1 , v 2 ) → a ( u ′ , v ′ 1 , v ′ 2 ). It giv es rise to an epiv alence C → Λ- mo d , where Λ is the path algebra of the quiv er • / / • ( ( 6 6 • . It is kno wn to b e wild. Therefore, T 8 is wild a s w ell.  MA TRIX PROBLEM S AND S T ABLE HOMOTOPY 31 Reference s [1] H. Bass. Algebraic K -theo ry . W. A. Be nja min, Inc. New Y or k – Amsterdam, 1968. [2] H.-J. Baues. Homotopy types. Handb o ok of Algebraic T op ology . Elsevier, Am- sterdam, 199 5, 1–7 2. [3] H.-J. Baues. A to ms in T op olo gy . J a hresb er. Dtsch. Math.–V er. 1 04 (20 02), 147–1 64. [4] H.-J. Baues and Y. A. Drozd. The homotopy cla ssification of ( n − 1)-connected ( n + 4)-dimensiona l p oly hedra with to rsion free ho mology . Expo . Math. 17 (1999), 161– 179. [5] H.-J. Baues and Y. A. Drozd. Repres en ta tion theory of homotopy types with at most tw o non-trivia l homotopy gro ups . Math. Pro c. Cambridge P hil. So c. 128 (2000), 283 –300. [6] H.-J. Baues and Y. A. Drozd. Indecomposa ble homotopy types with at most t wo non- tr ivial homo logy gro ups. Groups of Homotopy Self-Equiv alences a nd Related T opics. Contempora ry Mathematics, 274 (200 1), 3 9 –56. [7] H.-J. Baues and Y. A. Droz d. C la ssification of stable homotopy types with torsion-fr e e homology . T op ology 40 (2001), 789–82 1. [8] H. J. Baues and M. Hennes. The homotopy cla ssification of ( n − 1)-connected ( n + 3 )-dimensional p olyhedra, n ≥ 4. T o p olo gy , 3 0 (199 1), 3 7 3–408 . [9] V. M. Bondar enko. Repres ent a tions of bundles of semichained sets and their applications. St. Petersburg Math. J. 3 (1992), 973–9 96. [10] I. Bur ba n and Y u. Dro z d. Derived categories of no dal a lg ebras. J. Algebra 272 (2004), 46–9 4, arXiv:ma th/03070 60 . [11] S. C. Chang. Homotopy inv aria n ts a nd contin uous mappings. P ro c. R. So c. London 202 (1950 ), 2 53–26 3. [12] J . M. Cohen. Stable Homoto p y . Lec tur e Notes in Math. 1 65 (1970). [13] Y. A. Drozd. Ad` ele s and integral r e pr esentations. Izv. Ak a d. Nauk SSSR, Ser. Mat. 33 (1969 ), 1080–1 088. (Math. USSR, Izv. 3 (1969), 101 9–102 6.) [14] Y. A. Drozd. Reduction algorithm and representations of boxes and algebras. Comtes Rendue Math. Aca d. Sci. Ca nada, 23 (2001), 97 –125. [15] Y. A. Drozd. Ma trix problems and stable ho motopy types of p o lyhedra. Central Europ ean J. Math. 2 (2 0 04), 420 – 447, ar Xiv:ma th/04042 52 . [16] Y. A. Drozd. On cla s sification o f to r sion fr e e p oly he dr a. Preprint MPI 2005-99. Max-Plank -Institut f ¨ ur Mathematik, 2005. [17] O . Y. Drozd-Kor olev a. Representations o f deco rated bunc hes o f chains. Preprint. Kiev, 2 009. [18] R. Hartshorn. Residues and Duality . Lecture Notes in Math. 20 (1966). [19] H.-W. Henn. Classificatio n of p-lo cal low-dimensional spectr a . J. Pure Ap pl. Algebra 45 (1987 ), 45–71. [20] S.- T . Hu. Homotop y Theory . Academic Press. New Y or k – London, 19 59. [21] H. Jacobinski. Genera and decompos itions of lattices over o rders. Acta Math. 121 (1968), 1–2 9. [22] D. Puppe, On the structure of stable homotop y theory . Co llo quium on Alge- braic T o po logy . Aarhus Universitet Matematisk Institut, 1 962, 65–7 1. [23] R. M. Switzer. Algebraic T op olog y — Homotopy and Homolog y . Springe r - V er lag. Berlin – Heidelb erg – New Y or k, 1975. [24] H. T o da. Co mp os itio n Metho ds in the Homotopy Gr oups of Spheres . Ann. Math. Studies, V ol. 4 9, Princeton, 19 62. [25] J . H. C. Whitehea d. The homo topy type of a specia l kind of p olyhedron. Ann. So c. Polon. Math. 2 1 (1949), 17 6–186 . 32 YU RIY A. DROZD Institute of Ma thema tics, Na tional Academy of Sciences of Ukraine, Tereschenkivska 3, 01601 Kiev, Ukraine E-mail add re s s : dr ozd@im ath.ki ev.ua, y.a. drozd@ gmail.com URL : www. imath. kiev.ua/ ∼ drozd