Topological classification of the oriented cycles of linear mappings

We consider oriented cycles of linear mappings over the fields of real and complex numbers. the problem of their classification to within the homeomorphisms of spaces is reduced to the problem of classification of linear operators to within the homeomorphisms of spaces studied by N. Kuiper and J. Ro...

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Published in:	Український математичний журнал
Date:	2014
Main Authors:	Rybalkina, T.V., Sergeichuk, V.V.
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Cite this:	Topological classification of the oriented cycles of linear mappings / T.V. Rybalkina, V.V. Sergeichuk // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1414–1419. — Бібліогр.: 21 назв. — англ.

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author	Rybalkina, T.V. Sergeichuk, V.V.
author_facet	Rybalkina, T.V. Sergeichuk, V.V.
citation_txt	Topological classification of the oriented cycles of linear mappings / T.V. Rybalkina, V.V. Sergeichuk // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1414–1419. — Бібліогр.: 21 назв. — англ.
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description	We consider oriented cycles of linear mappings over the fields of real and complex numbers. the problem of their classification to within the homeomorphisms of spaces is reduced to the problem of classification of linear operators to within the homeomorphisms of spaces studied by N. Kuiper and J. Robbin in 1973. Розглядаються орiєнтованi цикли лінійних відображень над полями дійсних та комплексних чисел. Задача їхньої класифiкацiї з точністю до гомеоморфізмів просторів зводиться до задачі класифікації лінійних операторів з точністю до гомеоморфізмів просторів, яку вивчали Н. Койпер та Дж. Роббін у 1973 році.
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fulltext	UDC 515.126, 515.127 T. V. Rybalkina, V. V. Sergeichuk* (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv) TOPOLOGICAL CLASSIFICATION OF ORIENTED CYCLES OF LINEAR MAPPINGS ТОПОЛОГIЧНА КЛАСИФIКАЦIЯ ОРIЄНТОВАНИХ ЦИКЛIВ ЛIНIЙНИХ ВIДОБРАЖЕНЬ We consider oriented cycles of linear mappings over the fields of real and complex numbers. Тhe problem of their classification to within the homeomorphisms of spaces is reduced to the problem of classification of linear operators to within the homeomorphisms of spaces studied by N. Kuiper and J. Robbin in 1973. Розглядаються орiєнтованi цикли лiнiйних вiдображень над полями дiйсних та комплексних чисел. Задача їхньої класифiкацiї з точнiстю до гомеоморфiзмiв просторiв зводиться до задачi класифiкацiї лiнiйних операторiв з точнiс- тю до гомеоморфiзмiв просторiв, яку вивчали Н. Койпер та Дж. Роббiн у 1973 роцi. 1. Introduction. We consider the problem of topological classification of oriented cycles of linear mappings. Let A : V1 hh At A1 // V2 A2 // . . . Vt´1// At´2 At´1 // Vt (1) and B : W1 hh Bt B1 // W2 B2 // . . . Wt´1 // Bt´2 Bt´1 // Wt (2) be two oriented cycles of linear mappings of the same length t over a field F. We say that a system ϕ “ tϕi : Vi ÑWiu t i“1 of bijections transforms A to B if all squares in the diagram V1 ϕ1 �� jj At A1 // V2 ϕ2 �� A2 // . . . Vt´1 ϕt´1 �� // At´2 At´1 // Vt ϕt �� W1 jj Bt B1 // W2 B2 // . . . Wt´1 // Bt´2 Bt´1 // Wt (3) are commutative; that is, ϕ2A1 “ B1ϕ1, . . . , ϕtAt´1 “ Bt´1ϕt´1, ϕ1At “ Btϕt. (4) * V. V. Sergeichuk was supported in part by the Foundation for Research Support of the State of São Paulo (FAPESP), grant 2012/18139-2. c© T. V. RYBALKINA, V. V. SERGEICHUK, 2014 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 1407 1408 T. V. RYBALKINA, V. V. SERGEICHUK Definition 1. Let A and B be cycles of linear mappings of the form (1) and (2) over a field F. (i) A and B are isomorphic if there exists a system of linear bijections that transforms A to B. (ii) A and B are topologically equivalent if F “ C or R, Vi “ Fmi , Wi “ Fni for all i “ 1, . . . , t, and there exists a system of homeomorphisms1 that transforms A to B. The direct sum of cycles (1) and (2) is the cycle A‘ B : V1 ‘W1 jj At‘Bt A1‘B1 // V2 ‘W2 A2‘B2 // . . . At´1‘Bt´1// Vt ‘Wt . The vector dimA :“ pdimV1, . . . ,dimVtq is the dimension of A. A cycle A is indecomposable if its dimension is nonzero and A cannot be decomposed into a direct sum of cycles of smaller dimensions. A cycle A is regular if all A1, . . . , At are bijections, and singular otherwise. Each cycle A possesses a regularizing decomposition A “ Areg ‘A1 ‘ . . .‘Ar, (5) in which Areg is regular and all A1, . . . ,Ar are indecomposable singular. An algorithm that constructs a regularizing decomposition of a nonoriented cycle of linear mappings over C and uses only unitary transformations was given in [3]. The following theorem reduces the problem of topological classification of oriented cycles of linear mappings to the problem of topological classification of linear operators. Theorem 1. (a) Let F “ C or R, and let A : Fm1 ii At A1 // Fm2 A2 // . . . Fmt´1// At´2 At´1 // Fmt (6) and B : Fn1 hh Bt B1 // Fn2 B2 // . . . Fnt´1// Bt´2 Bt´1 // Fnt (7) be topologically equivalent. Let A “ Areg ‘A1 ‘ . . .‘Ar, B “ Breg ‘ B1 ‘ . . .‘ Bs (8) be their regularizing decompositions. Then their regular parts Areg and Breg are topologically equivalent, r “ s, and after a suitable renumbering their indecomposable singular summands Ai and Bi are isomorphic for all i “ 1, . . . , r. 1By [1] (Corollary 19.10) or [2] (Section 11) m1 “ n1, . . . ,mt “ nt. ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 TOPOLOGICAL CLASSIFICATION OF ORIENTED CYCLES OF LINEAR MAPPINGS 1409 (b) Each regular cycle A of the form (6) is isomorphic to the cycle A1 : Fm1 ii At...A2A1 1 // Fm2 1 // . . . Fmt´1//1 1 // Fmt . (9) If cycles (6) and (7) are regular, then they are topologically equivalent if and only if the linear operatorsAt . . . A2A1 andBt . . . B2B1 are topologically equivalent pas the cycles Fm1 ý At . . . A2A1 and Fn1 ý Bt . . . B2B1 of length 1q. Kuiper and Robbin [4, 5] gave a criterion for topological equivalence of linear operators over R without eigenvalues that are roots of 1. Budnitska [6] (Theorem 2.2) found a canonical form with respect to topological equivalence of linear operators over R and C without eigenvalues that are roots of 1. The problem of topological classification of linear operators with an eigenvalue that is a root of 1 was studied by Kuiper and Robbin [4, 5], Cappell and Shaneson [7 – 11], and Hsiang and Pardon [12]. The problem of topological classification of affine operators was studied in [6, 13 – 16]. The topological classifications of pairs of counter mappings V1ÝÑÐÝV2 (i.e., oriented cycles of length 2) and of chains of linear mappings were given in [17] and [18]. 2. Oriented cycles of linear mappings up to isomorphism. This section is not topological; we construct a regularizing decomposition of an oriented cycle of linear mappings over an arbitrary field F. A classification of cycles of length 1 (i.e., linear operators V ý) over any field is given by the Frobenius canonical form of a square matrix under similarity. The oriented cycles of length 2 (i.e., pairs of counter mappings V1ÝÑÐÝV2) are classified in [19, 20]. The classification of cycles of arbitrary length and with arbitrary orientation of its arrows is well known in the theory of representations of quivers; see [21] (Section 11.1). For each c P Z, we denote by rcs the natural number such that 1 ď rcs ď t, rcs ” c pmod tq. By the Jordan theorem, for each indecomposable singular cycle V ý A there exists a basis e1, . . . , en of V in which the matrix of A is a singular Jordan block. This means that the basis vectors form a Jordan chain e1 A ÝÑ e2 A ÝÑ e3 A ÝÑ . . . A ÝÑ en A ÝÑ 0. In the same manner, each indecomposable singular cycle A of an arbitrary length t also can be given by a chain ep Ap ÝÝÑ ep`1 Arp`1s ÝÝÝÝÑ ep`2 Arp`2s ÝÝÝÝÑ . . . Arq´1s ÝÝÝÝÑ eq Arqs ÝÝÑ 0 in which 1 ď p ď q ď t and for each l “ 1, 2, . . . , t the set tei\|i ” l pmod tqu is a basis of Vl; see [21] (Section 11.1). We say that this chain ends in Vrqs since eq P Vrqs. The number q ´ p is called the length of the chain. For example, the chain e4 // e5 rre6 // e7 // e8 // e9 // e10 rre11 // e12 // 0 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 1410 T. V. RYBALKINA, V. V. SERGEICHUK of length 8 gives an indecomposable singular cycle on the spaces V1 “ Fe6‘Fe11, V2 “ Fe7‘Fe12, V3 “ Fe8, V4 “ Fe4 ‘ Fe9, V5 “ Fe5 ‘ Fe10. Lemma 1. Let A : V1 hh At A1 // V2 A2 // . . . Vt´1// At´2 At´1 // Vt be an oriented cycle of linear mappings, and let (5) be its regularizing decomposition. (a) Write Âi :“ Ari`t´1s . . . Ari`1sAi : Vi Ñ Vi (10) and fix a natural number z such that Ṽi :“ Âz iVi “ Âz`1 i Vi for all i “ 1, . . . , t. Let Ã : Ṽ1 hh Ãt Ã1 // Ṽ2 Ã2 // . . . Ṽt´1// Ãt´2 Ãt´1 // Ṽt be the cycle formed by the restrictions Ãi : Ṽi Ñ Ṽri`1s of Ai : Vi Ñ Vri`1s. Then Areg “ Ã pand so the regular part is uniquely determined by Aq. (b) The numbers kij :“ dimKerpAri`js . . . Ari`1sAiq, i “ 1, . . . , t and j ě 0, determine the singular summands A1, . . . ,Ar of regularizing decomposition (5) up to isomorphism since the number nlj pl “ 1, . . . , t and j ě 0q of singular summands given by chains of length j that end in Vl can be calculated by the formula nlj “ krl´js,j ´ krl´js,j´1 ´ krl´j´1s,j`1 ` krl´j´1s,j (11) in which ki,´1 :“ 0. Proof. (a) Let (5) be a regularizing decomposition of A. Let Vi “ Vi,reg ‘ Vi1 ‘ . . .‘ Vir, i “ 1, . . . , t, be the corresponding decompositions of its vector spaces. Then Âz iVi,reg “ Vi,reg (since all linear mappings in Areg are bijections) and Âz iVi1 “ . . . “ Âz iVir “ 0. Hence Vi,reg “ Ṽi, and so Areg “ Ã. (b) Denote by σij :“ nij ` ni,j`1 ` ni,j`2 ` . . . the number of chains of length ě j that end in Vi. Clearly, ki0 “ σi0, ki1 “ σi0 ` σri`1s,1, . . . , and kij “ σi0 ` σri`1s,1 ` . . .` σri`js,j for each 1 ď i ď t and j ě 0. Therefore, σlj “ kij ´ ki,j´1, l :“ ri` js (recall that ki,´1 “ 0). This means that l ” i` j pmod tq, i ” l ´ j pmod tq, i “ rl ´ js, and so σlj “ krl´js,j ´ krl´js,j´1. We get nlj “ σlj ´ σl,j`1 “ krl´js,j ´ krl´js,j´1 ´ krl´j´1s,j`1 ` krl´j´1s,j . Lemma 1 is proved. ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 TOPOLOGICAL CLASSIFICATION OF ORIENTED CYCLES OF LINEAR MAPPINGS 1411 3. Proof of Theorem 1. In this section, F “ C or R. (a) Let A and B be cycles (6) and (7). Let them be topologically equivalent; that is, A is transformed to B by a system tϕi : Fmi Ñ Fniuti“1 of homeomorphisms. Let (8) be regularizing decompositions of A and B. First we prove that their regular parts Areg and Breg are topologically equivalent. In notation (10), Âi “ Ari`t´1s . . . Ari`1sAi, B̂i “ Bri`t´1s . . . Bri`1sBi. Let z be a natural number that satisfies both Âz iFmi “ Âz`1 i Fmi and B̂z i Fni “ B̂z`1 i Fni for all i “ 1, . . . , t. By (3), the diagram Fmi Âz // ϕi �� Fmi ϕi �� Fni B̂z // Fni is commutative. Then ϕi Im Âz i “ Im B̂z i for all i. Therefore, the restriction ϕ̂i : Im Âz i Ñ Im B̂z i is a homeomorphism. The system of homeomorphisms ϕ̂1, . . . , ϕ̂t transforms Ã to B̃, which are the regular parts of A and B by Lemma 1(a). Let us prove that r “ s, and, after a suitable renumbering, Ai and Bi are isomorphic for all i “ 1, . . . , r. Since all summands Ai and Bi with i ě 1 can be given by chains of basic vectors, it suffices to prove that nij “ n1ij for all i and j, where n1ij is the number of singular summands B1, . . . ,Bs in (8) given by chains of length j that end in the i th space Fni . Due to (11), it suffices to prove that the numbers kij are invariant with respect to topological equivalence. In the same manner as kij is constructed by A, we construct k1ij by B. Let us fix i and j and prove that kij “ k1ij . Write A :“ Ari`js . . . Ari`1sAi, B :“ Bri`js . . . Bri`1sBi, q :“ ri` j ` 1s and consider the commutative diagram Fmi A // ϕi �� Fmq ϕq �� Fni B // Fnq (12) which is a fragment of (3). We have kij “ dimKerA “ mi ´ dim ImA, k1ij “ ni ´ dim ImB. Because ϕi : Fmi Ñ Fni is a homeomorphism, mi “ ni (see [1], Corollary 19.10, or [2], Section 11). Since the diagram (12) is commutative, ϕqpImAq “ ImB. Hence, the vector spaces ImA and ImB are homeomorphic, and so dim ImA “ dim ImB, which proves kij “ k1ij . (b) Each regular cycle A of the form (6) is isomorphic to the cycle A1 of the form (9) since the diagram ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 1412 T. V. RYBALKINA, V. V. SERGEICHUK Fm1 1 �� jj At...A2A1 1 // Fm2 A1 �� 1 // Fm3 A2A1 �� Fm4 A3A2A1 �� //1 1 // . . . 1 // Fmt At´1...A2A1 �� Fm1 jj At A1 // Fm2 A2 // Fm3 Fm4//A3 A4 // . . . At´1 // Fmt (13) is commutative. Let A and B be regular cycles of the form (6) and (7). Let them be topologically equivalent; that is, A is transformed to B by a system ϕ “ pϕ1, . . . , ϕtq of homeomorphisms; see (3). By (4), ϕ1AtAt´1 . . . A1 “ BtϕtAt´1 . . . A1 “ BtBt´1ϕt´1At´2 . . . A1 “ . . . “ BtBt´1 . . . B1ϕ1, and so the cycles Fm1 ýAt . . . A2A1 and Fm1 ýBt . . . B2B1 are topologically equivalent via ϕ1. Conversely, let Fm1 ýAt . . . A2A1 and Fm1 ýBt . . . B2B1 be topologically equivalent via some homeomorphism ϕ1, and let A1 and B1 be constructed by A and B as in (9). Then A1 and B1 are topologically equivalent via the system of homeomorphisms ϕ “ pϕ1, ϕ1, . . . , ϕ1q. Let ε and δ be systems of linear bijections that transform A1 to A and B1 to B; see (13). Then A and B are topologically equivalent via the system of homeomorphisms δϕε´1. 1. Bredon G. E. Topology and geometry. – New York: Springer-Verlag, 1993. – 587 p. 2. McCleary J. A first course in topology: continuity and dimension. – Providence: Amer. Math. Soc., 2006. – 141 p. 3. Sergeichuk V. V. Computation of canonical matrices for chains and cycles of linear mappings // Linear Algebra and Appl. – 2004. – 376. – P. 235 – 263. 4. Robbin J. W. Topological conjugacy and structural stability for discrete dynamical systems // Bull. Amer. Math. Soc. – 1972. – 78. – P. 923 – 952. 5. Kuiper N. H., Robbin J. W. Topological classification of linear endomorphisms // Invent. Math.– 1973. – 19, № 2. – P. 83 – 106. 6. Budnitska T. Topological classification of affine operators on unitary and Euclidean spaces // Linear Algebra and Appl. – 2011. – 434. – P. 582 – 592. 7. Cappell S. E., Shaneson J. L. Linear algebra and topology // Bull. Amer. Math. Soc. (N. S.). – 1979. – 1. – P. 685 – 687. 8. Cappell S. E., Shaneson J. L. Nonlinear similarity of matrices // Bull. Amer. Math. Soc. (N. S.). – 1979. – 1. – P. 899 – 902. 9. Cappell S. E., Shaneson J. L. Non-linear similarity // Ann. Math. – 1981. – 113, № 2. – P. 315 – 355. 10. Cappell S. E., Shaneson J. L. Non-linear similarity and linear similarity are equivariant below dimension 6 // Contemp. Math. – 1999. – 231. – P. 59 – 66. 11. Cappell S. E., Shaneson J. L., Steinberger M., West J. E. Nonlinear similarity begins in dimension six // Amer. J. Math. – 1989. – 111. – P. 717 – 752. 12. Hsiang W. C., Pardon W. When are topologically equivalent orthogonal transformations linearly equivalent? // Invent. Math. – 1982. – 68, № 2. – P. 275 – 316. 13. Ephrämowitsch W. Topologische Klassifikation affiner Abbildungen der Ebene // Mat. Sb. – 1935. – 42, № 1. – P. 23 – 36. 14. Blanc J. Conjugacy classes of affine automorphisms of Kn and linear automorphisms of Pn in the Cremona groups // Manuscr. Math. – 2006. – 119, № 2. – P. 225 – 241. 15. Budnitska T. V. Classification of topological conjugate affine mappings // Ukr. Math. J. – 2009. – 61, № 2. – P. 164 – 170. 16. Budnitska T., Budnitska N. Classification of affine operators up to biregular conjugacy // Linear Algebra and Appl. – 2011. – 434. – P. 1195 – 1199. ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 TOPOLOGICAL CLASSIFICATION OF ORIENTED CYCLES OF LINEAR MAPPINGS 1413 17. Rybalkina T. Topological classification of pairs of counter linear maps // Mat. Stud. – 2013. – 39, № 1. – P. 21 – 28 (in Ukrainian). 18. Rybalkina T., Sergeichuk V. V. Topological classification of chains of linear mappings // Linear Algebra and Appl. – 2012. – 437. – P. 860 – 869. 19. Dobrovol’skaya N. M., Ponomarev V. A. A pair of counter-operators // Uspekhi Mat. Nauk. – 1965. – 20, № 6. – P. 80 – 86 (in Russian). 20. Horn R. A., Merino D. I. Contragredient equivalence: a canonical form and some applications // Linear Algebra and Appl. – 1995. – 214. – P. 43 – 92. 21. Gabriel P., Roiter A. V. Representations of finite-dimensional algebras // Encycl. Math. Sci. – Berlin: Springer-Verlag, 1992. – 73. – 177 p. Received 16.07.13 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1027-3190
language	English
last_indexed	2025-12-01T20:07:41Z
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spelling	Rybalkina, T.V. Sergeichuk, V.V. 2020-02-18T17:43:44Z 2020-02-18T17:43:44Z 2014 Topological classification of the oriented cycles of linear mappings / T.V. Rybalkina, V.V. Sergeichuk // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1414–1419. — Бібліогр.: 21 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166310 515.126 515.127 We consider oriented cycles of linear mappings over the fields of real and complex numbers. the problem of their classification to within the homeomorphisms of spaces is reduced to the problem of classification of linear operators to within the homeomorphisms of spaces studied by N. Kuiper and J. Robbin in 1973. Розглядаються орiєнтованi цикли лінійних відображень над полями дійсних та комплексних чисел. Задача їхньої класифiкацiї з точністю до гомеоморфізмів просторів зводиться до задачі класифікації лінійних операторів з точністю до гомеоморфізмів просторів, яку вивчали Н. Койпер та Дж. Роббін у 1973 році. V. V. Sergeichuk was supported in part by the Foundation for Research Support of the State of Sao Paulo (FAPESP), grant 2012/18139-2. en Інститут математики НАН України Український математичний журнал Статті Topological classification of the oriented cycles of linear mappings Топологічна класифікація орієнтованих циклів лінійних відображень Article published earlier
spellingShingle	Topological classification of the oriented cycles of linear mappings Rybalkina, T.V. Sergeichuk, V.V. Статті
title	Topological classification of the oriented cycles of linear mappings
title_alt	Топологічна класифікація орієнтованих циклів лінійних відображень
title_full	Topological classification of the oriented cycles of linear mappings
title_fullStr	Topological classification of the oriented cycles of linear mappings
title_full_unstemmed	Topological classification of the oriented cycles of linear mappings
title_short	Topological classification of the oriented cycles of linear mappings
title_sort	topological classification of the oriented cycles of linear mappings
topic	Статті
topic_facet	Статті
url	https://nasplib.isofts.kiev.ua/handle/123456789/166310
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Topological classification of the oriented cycles of linear mappings

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