(Đề tài NCKH) một số tính chất của toán tử đối đồng điều và ứng dụng

BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT THÀNH PHỐ HỒ CHÍ MINH CƠNG TRÌNH NGHIÊN CỨU KHOA HỌC CẤP TRƯỜNG MỘT SỐ TÍNH CHẤT CỦA TOÁN TỬ ĐỐI ĐỒNG ĐIỀU VÀ ỨNG DỤNG Mà SỐ: T2019- SKC006773 Tp Hồ Chí Minh, tháng 02/2020 B¸GI ODệCV TRìNG OT O I HC Sì PH M Kò THU T TH NH PHă H CH MINH B OC OT˚NGK T T I KH&CN C P TR×˝NG D NH CHO GI NG VI N TR MáTSăTNHCH TCếATO NT ăI ˙NG I U V ÙNG DƯNG M¢ sŁ: T2019-11GVT Chı nhiằm ã t i: ThS PHAN PHìèNG DUNG TP H CH MINH, 02/2020 TRìNG I HC Sì PH M Kò THU T TH NH PHă H CH MINH KHOA KHOA H¯C ÙNG DÖNG B OC OT˚NGK T T I KH&CN C P TR×˝NG D NH CHO GI NG VI N TR MáTSăTNHCH TCếATO NT ăI NG I U V NG DệNG M s: T2019-11GVT Ch nhiằm Th nh viản ã t i: PHAN PHìèNG DUNG ã t i: NGUY N KH C T N TP H˙ CH MINH, 02/2020 Möc löc PH NM— U Mºt sŁ ki‚n thøc chu'n bà 1.1 ⁄i sŁ v Łi ⁄i sŁ ph¥n b“c- ⁄i sŁ H 1.2 ⁄i sŁ tenxì 1.3 ⁄i sŁ Łi xøng 1.4 ⁄i sŁ ngo i ToĂn tò i ỗng iãu 2.1 ⁄i sŁ Steenrod modulo 2.2 C§u tróc ⁄i sŁ Hopf cıa ⁄i sŁ St 2.3 C§u tróc A-mỉ un cıa H (RP 2.4 2.5 ⁄i sŁ Steenrod A(p), vỵi p > B§t bi‚n Dickson-Mịi 2.6 B i to¡n hit Łi vỵi ⁄i sŁ Steenrod 2.6.1 2.6.2 2.6.3 Ùng döng cıa b i toĂn hit cho ỗng cĐu chuyn i s ca Singer 3.1 GiÊ thuyt ca Singer vã ỗng cĐ 3.2 K‚t qu£ ch‰nh K‚t lu“n i I H¯C Sì PH M Kò THU T CáNG HO X HáI CHÕ NGH A VI T NAM ºc l“p - Tü - Hnh phúc TH NH PHă H CH MINH KHOA KHOA H¯C ÙNG DÖNG Tp HCM, ng y 20 th¡ng 02 n«m 2020 TH˘NG TIN K T QU NGHI N CU Thổng tin chung: - Tản ã t i: "Mt s tnh chĐt ca toĂn tò i ỗng iãu v ứng dửng" - M s: T2019-11GVT - Ch nhi»m: ThS Phan Ph÷ìng Dung - Cì quan chı tr…: i hồc Sữ phm K thut Th nh ph Hỗ Ch‰ Minh - Thíi gian thüc hi»n: 12 th¡ng Mửc tiảu: XĂc nh tữớng minh cỡ s chĐp nhn ÷ỉc cıa ⁄i sŁ a thøc n«m bi‚n Ùng dưng k‚t qu£ n y ” nghi¶n cøu ki”m ành gi£ thuyt ca Singer vã ỗng cĐu chuyn i s Tnh mợi v sĂng to: Trong ã t i, nhõm nghiản cứu  ữa cĂc kt quÊ mợi: r+2 ành l‰ Cho n = 11:2 5, vỵi r l mt s nguyản dữỡng bĐt k Khi tỗn ti 134190 ỡn thức chĐp nhn ữổc bc n P6, vợi r > Do dim(F2 A P )n â, â, = 134190 K‚t qu£ nghi¶n cøu: Chóng tỉi nghiản cứu b i toĂn hit i vợi i s a thức ữổc xt nhữ mt mổ un trản i sŁ Steenrod t⁄i mºt sŁ b“c Sß dưng k‚t qu£ n y, chóng tỉi ch¿ r‹ng gi£ thuy‚t cıa Singer vã ỗng cĐu chuyn i s l úng tr÷íng hỉp n y S£n ph'm: 01 b i b¡o [1] P P Dung, H N Ly, N K Tin, "Dimension result for the polynomial algebra of six variables as a module over Steenrod algebra in some degrees", ii Journal of Technical Education Science, 6-pages, 2020 (accepted) Hi»u qu£, phữỡng thức chuyn giao kt quÊ nghiản cứu v khÊ nông Ăp dửng: Cổng b kt quÊ nghiản cứu vỵi mưc ‰ch håc thu“t, phưc vư tham kh£o cho giĂo dửc o to v nghiản cứu cỡ bÊn Trững ỡn v Ch nhiằm ãti Phan Phữỡng Dung Th nh vi¶n: Nguy„n Kh›c T‰n iii INFORMATION ON RESEARCH RESULTS General information: Project title: "Some properties of the cohomology operations and its appli-cations" Code number: T2019-11GVT Coordinator: Phan Phuong Dung Implementing institution: University of Technology and Education, Ho Chi Minh City, Vietnam Duration: from January, 2019 to December, 2019 Objective(s): We explicitly determine the hit problem for the case of five variables in terms of the admissible monomials Applying the results for the hit problem to study and verify the Singer conjecture for the algebraic transfer Creativeness and innovativeness: This is a new contribution of authors ( see [10] and [49]) Research results: We study the hit problem for the polynomial algebra as a module over Steen-rod algebra in some degrees Using this results, we show that Singer’s conjecture for the algebraic transfer is true in this case Products: 01 paper [1] P P Dung, H N Ly, N K Tin, "Dimension result for the polynomial algebra of six variables as a module over Steenrod algebra in some degrees", Journal of Technical Education Science, 6-pages, 2020 (accepted) Effects, transfer alternatives of reserach results and applicability: The publishment of the research results is for academic purpose, utilized as a reference for education-training and fundamental research iv PH NM— U T˚NG QUAN T NH H NH NGHI N CU V MệC TI U CếA T I Lỵ thuy‚t b§t bi‚n modula cıa nhâm tuy‚n t‰nh tŒng qu¡t ữổc ã xữợng bi Dickson v o nhng nôm 1910 v thp niản n y nõ  phĂt trin mt cĂch mnh m vợi tữ cĂch l mt ng nh i s thun túy Tuy nhiản, mÂi n nhng nôm 1970 Mũi [17] phĂt trin thảm cho mt sŁ nhâm kh¡c cıa nhâm tuy‚n t ‰nh tŒng qu¡t v ¡p dưng nâ ” nghi¶n cøu c¡c ⁄i s i ỗng iãu ca cĂc nhõm i xứng th lỵ thuyt n y mợi tr th nh mt cổng cư hœu hi»u Tỉpỉ ⁄i sŁ; °c bi»t l thới gian gn Ơy nõ ữổc sò dửng nghiản cứu cĂc toĂn tò ỗng iãu v i ỗng iãu k d; phƠn tch n nh cĂc khổng gian ph¥n lo⁄i Cho X l mºt khỉng gian tỉpỉ, kỵ hiằu H (X; F2) i ỗng iãu k d ca X vợi hằ s trản trữớng F2 cõ phn tò CĂc h m tò ỗng iãu v i ỗng iãu ký d l cĂc bĐt bin ỗng luƠn ữổc sò dửng nghiản cứu mt nhng b i to¡n trång t¥m cıa Tỉpỉ ⁄i sŁ l b i toĂn phƠn loi kiu ỗng luƠn ca cĂc khổng gian tổpổ Tuy nhiản, nhiãu trữớng hổp, bĐt bin n y ch÷a ı m⁄nh ” gi£i quy‚t b i to¡n nâi tr¶n Mºt nhœng cỉng cư l m tinh t i ỗng iãu l toĂn tò i ỗng iãu ữổc xƠy dỹng nôm 1947 bi Steenrod v thữớng ữổc gồi l toĂn tò k Steenrod, Sq ; vợi Sqk : H (X; F2)! H +k (X; F2) t¡c ng tỹ nhiản lản i ỗng iãu k d modulo cıa khỉng gian tỉpỉ X vỵi k l sŁ nguyản khổng Ơm CĂc toĂn tò n y sau õ ữổc Thom v Wu sò dửng nghiản cứu cĂc lợp c trững ca phƠn thợ vectỡ v nhanh chõng trð th nh mºt nhœng cỉng cư h ng ƒu cıa Tỉpỉ ⁄i sŁ C§u tróc cıa t“p hỉp cĂc toĂn tò i ỗng iãu ữổc l m rê bði Serre v o n«m 1952 Serre chøng minh r‹ng, vợi php cng thổng thữớng v php hổp th nh cĂc Ănh x, cĂc toĂn tò Steenrod sinh tĐt cÊ cĂc toĂn tò i ỗng iãu n nh i s cĂc toĂn tò i ỗng iãu n nh vợi hằ s trản trữớng F ữổc gồi l i sŁ Steenrod modulo v th÷íng ÷ỉc k‰ hi»u l A: Nh÷ v“y, ⁄i sŁ Steenrod câ th” ÷ỉc ành nghắa mt cĂch thun túy i s i nhữ l th÷ìng cıa F2- ⁄i sŁ k‚t hỉp tü sinh bi cĂc kỵ hiằu Sq (vợi i l s nguyản khổng Ơm) chia cho i ảan hai pha sinh bði h» thøc Sq = v c¡c quan h» Adem SqaSqb = ⁄i sŁ A câ mºt c§u tróc phƠn bc tỹ nhiản xĂc nh bi deg(Sqi1 Sqi2 : : : Sqik ) = i1 + i2 : : : + ik vỵi måi i1; i2; : : : ; ik > 0: Hìn nœa, nâ l mºt ⁄i s phƠn bc cõ b sung, nghắa l cõ mt to n cĐu F2- i s phƠn bc tỹ nhiản : A! 0: L mt hổp cĂc toĂn tò i ỗng iãu, A tĂc ng mt cĂch tỹ nhiản lản i ỗng iãu ca mồi khổng gian tổpổ Do õ, i ỗng i•u cıa c¡c khỉng gian tỉpỉ khỉng ch¿ l F2- ⁄i sŁ m cỈn l mºt A-mỉ un Mºt nhœng b i to¡n m chóng tỉi quan t¥m l b i to¡n t…m t“p sinh cüc ti”u cıa ⁄i s a thức Pk ữổc xt nhữ mổ un trản ⁄i sŁ Steenrod A B i to¡n n y ÷ỉc gåi l b i to¡n hit Łi vỵi ⁄i sŁ Steenrod N‚u x†t F nh÷ mºt A-mỉ un tƒm thữớng th b i toĂn hit tữỡng ữỡng vợi b i to¡n t…m mºt cì sð cıa F 2khỉng gian v†ctì ph¥n b“c F A + â A l i ảan ca A sinh bi tĐt cÊ cĂc toĂn tò Steenrod bc dữỡng B i toĂn n y ÷ỉc nghi¶n cøu ƒu ti¶n bði Peterson [27, 28], Singer [36], Wood [55], Priddy [30] nhng ngữới  ch¿ mŁi li¶n h» cıa b i to¡n hit vỵi mºt sŁ b i to¡n cŒ i”n lỵ thuyt ỗng luƠn nhữ lỵ thuyt ỗng biản ca cĂc a tp, lỵ thuyt biu din modular ca cĂc nhõm tuyn tnh, dÂy ph Adams i vợi ỗng luƠn Œn ành cıa m°t cƒu v b i to¡n ph¥n t‰ch Œn ành c¡c khỉng gian ph¥n lo⁄i cıa nhâm hu hn Trong [27], Peterson  ữa giÊ thuyt rng, nhữ mt mổ un trản i s Steenrod, i sŁ a thøc Pk ÷ỉc sinh bði c¡c ìn thøc b“c n thäa m¢n (n + k) k; â (n) l sŁ h» sŁ khai tri”n nh phƠn ca n v chứng minh iãu n y vợi k 2: GiÊ thuyt n y ữổc Wood [55] chøng minh mºt c¡ch tŒng qu¡t v o n«m 1989 Ơy l mt cổng cử cỡ bÊn i vợi b i to¡n x¡c ành t“p sinh cüc ti”u cıa Amỉ un Pk: Sau â k‚t qu£ n y ÷ỉc ph¡t tri”n xa hìn bði Singer [36] v Silverman [33, 34] n nay, tch tenxỡ F2 APk  ữổc xĂc nh tữớng minh vợi k = 1; bi Peterson, vợi k = bi Kameko [21] Trữớng hổp k = ÷ỉc x¡c ành ho n to n bði Sum [41, 43] Trong tr÷íng hỉp tŒng qu¡t t⁄i mºt sŁ d⁄ng b“c n o â, b i to¡n ÷ỉc sỹ quan tƠm nghiản cứu bi nhiãu tĂc giÊ v ngo i nữợc (chflng hn nhữ: Boardman [2], Bruner-H -H÷ng [3], Carlisle-Wood [4], CrabbHubbuck [5], Giambalvo-Peterson [11], Janfada-Wood [19, 20], Mothebe [23], Nam [25], Repka-Selick [32], Silverman [33], Silverman-Singer [35], Singer [37], Sum [39, 40, 43], Walker-Wood [52, 53, 54], Wood [55, 56] v mºt sŁ t¡c gi£ kh¡c) Tuy nhiản, cĂc kt quÊ t ữổc vÔn cặn hn ch, cÊ trữớng hổp k = vợi sü hØ trỉ cıa m¡y t‰nh i»n tß Mºt nhœng øng döng quan trång cıa b i to¡n hit l sò dửng nõ viằc nghiản cứu mt ỗng cĐu ữổc thit lp bi Singer v o nôm 1989 Trong [36], Singer nh nghắa hng k : ỗng cĐu chuy”n ’k : TorAk;k+n(F2; F2) ! (F2 A Pk) GL k n ⁄i sŁ ; Pk)GLnk l khỉng gian v†ctì ca F A Pk gỗm tĐt cÊ cĂc lợp bc n bĐt bin i vợi tĂc ng ca nhâm tuy‚n t‰nh tŒng qu¡t GLk: â (F2 A Chuyn qua i ngÔu, ta cụng ữổc mt ỗng cĐu i s v cụng ữổc gồi l ỗng cĐu chuyn ⁄i sŁ cıa Singer T rk := (’k) : F2 GLk P Hd(H ((RP 1)k))! Ext k;k +d A (F2; F2): Theorem (Tin-Sum[17]) Let d be an a = α (a ).20 + α1 (a ).21 + α2 (a).22 + arbitrary non-negative integer Then (Sq*0 )s −t : (QPk )k for αi (a )∈ F2 and i ≥ ( Let x = x1a x2a xka ∈ Pk Denote τ j (x = a j , with ≤ j ≤ k Set is an isomorphism of GLk -modules for every s≥t )= Here {j ∈ k: ( t where ς (n) calculating the hit problem and the GLk invariant of polynomial algebra For now on, the tensor product F2 ⊗A Pk was explicitly calculated by Peterson [4] for k =1; , by Kameko [3] for k = and by Sum [9] for k = However, the problem is still unsolved with k ≥ , even in the case of k = with the help of computers The main result of the paper is the following r +2 Main Theorem Let n = 11.2 −5 , with r an arbitrary positive integer Then, there exist exactly 134190 admissible monomials of degree n in P6 , for r > Consequently, )n =134190 In Section 2, we recall some needed information on the admissible monomials and hit monomials in Pk The proof of Main Theorem is presented in Section ≥0 ) = (ω1 (x ),ω2 (x ),ω3 (x), )  ( x ) = (τ (x ),τ (x ), ,τk (x)) ( x m , with m an odd integer dim (F2 ⊗ A P6 ) = 0} for t Following Kameko [3], we define two sequences associated with x by that (n) α t (τ j ( x) ) ≥0.t t n = 2ς k PRELIMINARIES We first recall some results from Kameko [3], which will be used in the next section Notation 2.1 We and where k ωi (x ) = ∑α i −1 (τ j (x )) = degXℑ j =1 i−1 ( x) , i ≥1 The sequence ω( x) is called the weight vector of x The sets of all the weight vectors and the exponent vectors are given the left lexicographical order Definition 2.2 Let x, y be the monomials in Pk We say that x < y if and only if one of the following holds: (1) ω ( x ) < ω ( y) (2) ω ( x ) = ω ( y) and σ ( x ) < σ ( y) Here, the order on the set of sequences of nonnegative integers is the lexicographical one Let f , g be homogeneous polynomials of the same degree in Pk We denote f ≡ g if and only if f − g ∈ A+ Pk If f ≡ then f is called hit Definition 2.3 A monomial x is said to be inadmissible if there exist the monomials y1 , y , yt such that yi < x, i =1, 2, , t and t Let dyadic expansion of a non-negative integer a That means x− ∑ yi ∈ A+ Pk i=1 A monomial x is said to be admissible if x it is not inadmissible Obviously, the set of all admissible monomials in Pk is a generators of Pk For later use, we set  Pk =  x = x1a1 x2a2  f i (x j )=  x  j +1 ,  Then, it can be easily seen that if B is a minimal set of generators for A-module Pk −1 in degree n, then f (B )=k f i (B) is a i=1 minimal set of generators for A-module Pk0 in degree n  Pk+ =  x = x1a1 x2a2  It is easy to see that Po Based on the results in [13], we see submodules of Pk Furthermore, we have the following that B4 (39) = 225 and therefore Proposition 2.4 We have a direct summand decomposition of the F2 -vector spaces F2 ⊗ A Pk = ( F2 ⊗ A Pk0 )⊕ ( F2 ⊗A Pk+ ) in Pk , we and B5 (17) = 566 From the above represented results, one get B5 (39) = 2130 For a polynomial f denote by f  ] the class in F2 ⊗A Pk Moreover, Let n ∈ , we recall that the function by f  : → is given by: µ (0) = and PROOF OF MAIN THEOREM From now on, we will denote by Bk (n) the set of all admissible monomials of degree n in Pk We first recall a result in [6] on the dimension of the vector space (QP5 follows: )39 as   (n ) =  m ∈ : n = Sq*0 )( k , m) : (QPk )2m +k → (QPk Here α (n) denotes the number of ones in dyadic expansion of n By a direct computation, we see µ (39) = 3, α (39 + µ(39)) = = µ (39) )m we have Hence, ( B6 (11.2r +2 − 5) for any r > )39 (QP50 )39 For ≤ i ≤ k , define f i : Pk −1 → Pk of algebras by substituting − 1), di >  = min{m ∈ : α (n + m ) ≤ m} is an epimorphism of GLk -modules, and using Proposition 2.4, we get QP i i=1 Since the squaring operation   ∑m (2d And therefore, there exist exactly 134190 admissible monomials 11.2 r +2 −5 in P6 , for any integer r > Main Theorem is proved REMARK One of the major applications of hit problem is in surveying a homomorphism introduced by Singer in 1989 Singer [7] defined the algebraic transfer, which is a homomorphism ϕk : TorkA;k +n (F2 , F2 ) → ( F2 ⊗A Pk )GLn k from the homology of the Steenrod algebra to the the subspace of F2 ⊗A Pk consisting of all the GLk -invariant classes of degree n It is a useful tool in describing the homology groups of the Steenrod algebra, A Tork ;k +n (F2 , F2 ) Singer has indicated the importance of the algebraic transfer by showing that ϕk is an isomorphism with k =1; and at some other degrees with k = 3; but he also disproved this for ϕ5 at degree and then gave the following conjecture It could be seen from the work of Singer the meaning and necessity of the hit problem In 1991, Boardman confirmed this again by using the modular representation theory of linear groups to show that ϕ3 is an isomorphism Recently, Hung and his collaborators have completely determined the image Tr4 , here Tr4 is dual to ϕ4 Furthermore, Hung proved in [2] that for any k ≥ 4, ϕk is not an isomorphism in infinitely many degrees However, it has not been known whether the algebraic transfer fails to be a monomorphism or fails to be an epimorphism Therefore, Singer's conjecture is still open for k ≥ Then, the results of hit problem are used to verify Singer's conjecture for the algebraic transfer ACKNOWLEDGMENT We would like to thank Ho Chi Minh City University of Technology and Education for supporting this work Conjecture (Singer [7]) The algebraic transfer ϕk is an epimorphism for any k ≥ REFERENCES [1] J M Boardman, Modular representations on the homology of power of real projective space, Contemp Math., 146, pp 49-70, 1993 [2] N H V Hung, The cohomology of the Steenrod algebra and representations of the general linear groups, Trans Amer Math Soc., 357, pp 4065-4089, 2005 [3] M Kameko, Products of projective spaces as Steenrod modules, PhD Thesis, The Johns Hopkins University, ProQuest LLC, Ann Arbor, MI, 29 pp., 1990 [4] [5] F P Peterson, G Abstracts Amer S Priddy, On characterizing summands in the classifying space of a group I, Amer J Math., 112, pp 737-748, 1990 [6] D [7] W M Singer, The transfer in homological algebra, Math Zeit., 202, pp 493-523, 1989 [8] N E Steenrod and D B A Epstein, Cohomology operations, Annals of Mathematics Studies 50, Princeton University Press, New Jersey, 1962 [9] N Sum, The hit problem for the polynomial algebra of four variables, 240 pp.; available online at: http://arxiv.org/abs/1412.1709 [10] N Sum, The negative answer to Kameko's conjecture on the hit problem, C R Acad Sci Paris Ser I, 348, pp 669-672, 2010 [11] N Sum, The negative answer to Kameko's conjecture on the hit problem, Adv Math., 225, pp 2365-2390, 2010 [12] N Sum, On the hit problem for the polynomial algebra, C R Acad Sci Paris Ser I, 351, pp 565-568, 2013 [13] N Sum, On the Peterson hit problem, Adv Math., 274, pp 432-489, 2015 [14] N Sum and N K Tin, Some results on the fifth Singer transfer, East-West J Math., 17 (1), pp 70-84, 2015 [15] N K Tin, The admissible monomial basis for the polynomial algebra of five variables in degree eleven, Journal of Science, Quy Nhơn University, (3), pp 81-89, 2012 [16] N K Tin, The admissible monomial basis for the polynomial algebra of five variables in degree s +1 + s −5 , East-West J Math, 16 (1), pp 34-46, 2014 [17] N K Tin and N Sum, Kameko's homomorphism and the algebraic transfer, C R Acad Sci Paris Ser I, 354, pp 940-943, 2016 [18] N K Tin, On Singer's conjecture for the fifth algebraic transfer, Arxiv., 25-pages; available online at http://arxiv.org/abs/1609.02250 [19] N K Tin, The admissible monomial basis for the polynomial algebra as a module over Steenrod algebra in some degrees, Commun Korean Math Soc., 9-pages (Preprint-2019) [20] R M W Wood, Steenrod squares of polynomials and the Peterson conjecture, Math Proc Camb Phil Soc., 105, pp 307-309, 1989 Corresponding Author: Nguyen Khac Tin Ho Chi Minh City University of Technology and Education, Vietnam Email: tinnk@hcmute.edu.vn ... i(y) ỗng cĐu Bockstein tữỡng ứng vợi dÂy khợp cĂc nhõm hằ s â (x) = 2x; (y) = y mod 2: 11 CĐu trúc ca hổp cĂc toĂn tò i ỗng iãu ữổc l m rê bi Serre v o nôm 1952 Serre chứng minh rng, vợi php cng... kh¡c cıa nhâm tuy‚n t ‰nh tŒng qu¡t v ¡p dưng nâ ” nghi¶n cøu c¡c i s i ỗng iãu ca cĂc nhõm i xứng th lỵ thuyt n y mợi tr th nh mt cỉng cư hœu hi»u Tỉpỉ ⁄i sŁ; °c bi»t l thới gian gn Ơy nõ ữổc... Tỉpỉ ⁄i sŁ C§u tróc cıa t“p hổp cĂc toĂn tò i ỗng iãu ữổc l m rê bi Serre v o nôm 1952 Serre chứng minh rng, vợi php cng thổng thữớng v php hổp th nh c¡c ¡nh x⁄, c¡c to¡n tß Steenrod sinh tĐt