Topological classification of oriented cycles of linear mappings

0.18 MB·

by  Tetiana Rybalkina

Topological classification of oriented cycles of
4
linear mappings˚
1
0
2

n Tetiana Rybalkina Vladimir V. Sergeichuk
a
J rybalkina [email protected] [email protected]

1
Institute of Mathematics, Tereshchenkivska 3, Kiev, Ukraine
1

]
T
R
. Abstract
h
t
a
We consider oriented cycles
m
[
1 Fm1 hh A1 // Fm2 A2 // ... At´2// Fmt´1 At´1 //Fmt
v At
0
5 of linear mappings over F “ C or R, and reduce the problem of their
5
2 classification up to homeomorphisms in the spaces Fm1,...,Fmt to
. the case t “ 1, which was studied by N.H. Kuiper and J.W. Robbin
1
0 [Invent. Math. 19 (no.2) (1973) 83–106] and by other authors.
4
1 Keywords: Oriented cycles of linear mappings; Topological equiva-
: lence
v
i 2000 MSC: 15A21; 37C15
X
r
a
1 Introduction
We consider the problem of topological classification of oriented cycles of
linear mappings.
˚PrintedinUkrainian Math. J.,2014. Thesecondauthorwassupportedinpartbythe
FoundationforResearchSupportofthe StateofSa˜oPaulo(FAPESP),grant2012/18139-
2.
1
Let
A : V1 hh A1 // V2 A2 // ... At´2// Vt´1 At´1 // Vt (1)
At
and
B : W1 hh B1 //W2 B2 // ... Bt´2// Wt´1 Bt´1 // Wt (2)
Bt
be two oriented cycles of linear mappings of the same length t over a field F.
We say that a system ϕ “ tϕ : V Ñ W ut of bijections transforms A to B
i i i i“1
if all squares in the diagram
V1 jj A1 // V2 A2 // ... At´2 //Vt´1 At´1 //Vt
At
ϕ1 ϕ2 ϕt´1 ϕt (3)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
W1 jj B1 //W2 B2 // ... Bt´2// Wt´1 Bt´1 //Wt
Bt
are commutative; that is,
ϕ A “ B ϕ , ..., ϕ A “ B ϕ , ϕ A “ B ϕ . (4)
2 1 1 1 t t´1 t´1 t´1 1 t t t
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,
V “ Fmi, W “ Fni for all i “ 1,...,t,
i i
and there exists a system of homeomorphisms1 that transforms A to
B.
1By [2, Corollary 19.10] or [17, Section 11], m1 “n1,...,mt “nt.
2
The direct sum of cycles (1) and (2) is the cycle
A‘B : V ‘W A1‘B1 // V ‘W A2‘B2 // ... At´1‘Bt´1// V ‘W
1 1jj 2 2 t t
At‘Bt
The vector dimA :“ pdimV ,...,dimV q is the dimension of A. A cycle
1 t
A is indecomposable if its dimension is nonzero and A cannot be decomposed
into a direct sum of cycles of smaller dimensions.
AcycleAisregular ifallA ,...,A arebijections, andsingular otherwise.
1 t
Each cycle A possesses a regularizing decomposition
A “ A ‘A ‘¨¨¨‘A , (5)
reg 1 r
in which A is regular and all A ,...,A are indecomposable singular. An
reg 1 r
algorithmthatconstructs aregularizingdecomposition ofanonorientedcycle
of linear mappings over C and uses only unitary transformations was given
in [21].
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 A1 //Fm2 A2 // ... At´2// Fmt´1 At´1 //Fmt (6)
At
and
B : Fn1 hh B1 // Fn2 B2 // ... Bt´2// Fnt´1 Bt´1 //Fnt (7)
Bt
be topologically equivalent. Let
A “ A ‘A ‘¨¨¨‘A , B “ B ‘B ‘¨¨¨‘B (8)
reg 1 r reg 1 s
be their regularizing decompositions. Then their regular parts A and B
reg reg
are topologically equivalent, r “ s, and after a suitable renumbering their
indecomposable singular summands A and B are isomorphic for all i “
i i
1,...,r.
3
(b) Each regular cycle A of the form (6) is isomorphic to the cycle
A1 : Fm1 ii 1 //Fm2 1 // ... 1 // Fmt´1 1 //Fmt (9)
At¨¨¨A2A1
If cycles (6) and (7) are regular, then they are topologically equivalent if
and only if the linear operators A ¨¨¨A A and B ¨¨¨B B are topologically
t 2 1 t 2 1
equivalent pas the cycles Fm1ýA ¨¨¨A A and Fn1ýB ¨¨¨B B of length
t 2 1 t 2 1
1q.
Kuiper and Robbin [18, 16] gave a criterion for topological equivalence of
linear operators over R without eigenvalues that are roots of 1. Budnitska [4,
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 [18, 16], Cappell
and Shaneson [6, 7, 8, 9, 10], and Hsiang and Pardon [15]. The problem of
topologicalclassification ofaffineoperatorswasstudied in[4, 12, 1, 3, 5]. The
topological classifications of pairs of counter mappins V ÝÐÑÝV (i.e., oriented
1 2
cycles of length 2) and of chains of linear mappings were given in [19] and
[20].
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 mappins
V ÝÐÑÝV ) are classified in [11, 14]. The classification of cycles of arbitrary
1 2
lengthandwitharbitraryorientationofitsarrows iswell known inthetheory
of representations of quivers; see [13, Section 11.1].
For each c P Z, we denote by rcs the natural number such that
1 ď rcs ď t, rcs ” c mod t.
4
By the Jordan theorem, for each indecomposable singular cycle V ýA
there exists a basis e ,...,e of V in which the matrix of A is a singular
1 n
Jordan block. This means that the basis vectors form a Jordan chain
A A A A A
e ÝÑ e ÝÑ e ÝÑ ¨¨¨ ÝÑ e ÝÑ 0.
1 2 3 n
Inthesamemanner, eachindecomposable singularcycleAofanarbitrary
length t also can be given by a chain
e ÝAÑp e ÝAÝrÝp`ÝÑ1s e ÝAÝrÝp`ÝÑ2s ¨¨¨ ÝAÝrÝq´ÝÑ1s e ÝAÝrÑqs 0
p p`1 p`2 q
in which 1 ď p ď q ď t and for each l “ 1,2,...,t the set te |i ” l mod tu is
i
a basis of V ; see [13, Section 11.1]. We say that this chain ends in V since
l rqs
e P V . The number q ´p is called the length of the chain.
q rqs
For example, the chain
ee6 rr❞rr❡❞❡❞❡//❞//e❡e❞❡7❞❡❞❡❞❡❞❡❞//❡//e❞0❡❞8❡❞❡❞❡❞❡//❞❡ee❞❡49❞❡❞❡❞❡❞////ee150
11 12
of length 8 gives an indecomposable singular cycle on the spaces V “ Fe ‘
1 6
Fe , V “ Fe ‘Fe , V “ Fe , V “ Fe ‘Fe , V “ Fe ‘Fe .
11 2 7 12 3 8 4 4 9 5 5 10
Lemma 1. Let
A : V1 hh A1 //V2 A2 // ... At´2 //Vt´1 At´1 // Vt
At
be an oriented cycle of linear mappings, and let (5) be its regularizing decom-
position.
(a) Write
Aˆ :“ A ¨¨¨A A : V Ñ V (10)
i ri`t´1s ri`1s i i i
and fix a natural number z such that
V˜ :“ AˆzV “ Aˆz`1V for all i “ 1,...,t.
i i i i i
Let
A˜: V˜1 hh A˜1 // V˜2 A˜2 // ... A˜t´2 //V˜t´1 A˜t´1 // V˜t
A˜t
5
be the cycle formed by the restrictions A˜ : V˜ Ñ V˜ of A : V Ñ
i i ri`1s i i
V . Then A “ A˜ pand so the regular part is uniquely determined
ri`1s reg
by Aq.
(b) The numbers
k :“ dimKerpA ¨¨¨A A q, i “ 1,...,t and j ě 0,
ij ri`js ri`1s i
determine the singular summands A ,...,A of regularizing decompo-
1 r
sition (5) up to isomorphism since the number n pl “ 1,...,t and
lj
j ě 0q of singular summands given by chains of length j that end in V
l
can be calculated by the formula
n “ k ´k ´k `k (11)
lj rl´js,j rl´js,j´1 rl´j´1s,j`1 rl´j´1s,j
in which k :“ 0.
i,´1
Proof. (a) Let (5) be a regularizing decomposition of A. Let
V “ V ‘V ‘¨¨¨‘V , i “ 1,...,t,
i i,reg i1 ir
be the corresponding decompositions of its vector spaces. Then AˆzV “
i i,reg
V (since all linear mappings in A are bijections) and AˆzV “ ¨¨¨ “
i,reg reg i i1
AˆzV “ 0. Hence V “ V˜, and so A “ A˜.
i ir i,reg i reg
(b) Denote by
σ :“ n `n `n `¨¨¨
ij ij i,j`1 i,j`2
the number of chains of length ě j that end in V . Clearly, k “ σ ,
i i0 i0
k “ σ `σ , ..., and
i1 i0 ri`1s,1
k “ σ `σ `¨¨¨`σ
ij i0 ri`1s,1 ri`js,j
for each 1 ď i ď t and j ě 0. Therefore,
σ “ k ´k , l :“ ri`js
lj ij i,j´1
(recall that k “ 0). This means that l ” i ` j mod t, i ” l ´ j mod t,
i,´1
i “ rl´js, and so
σ “ k ´k .
lj rl´js,j rl´js,j´1
We get
n “ σ ´σ “ k ´k ´k `k .
lj lj l,j`1 rl´js,j rl´js,j´1 rl´j´1s,j`1 rl´j´1s,j
6
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 equiv-
alent; that is, A is transformed to B by a system tϕ : Fmi Ñ Fniut of
i i“1
homeomorphisms. Let (8) be regularizing decompositions of A and B.
First we prove that their regular parts A and B are topologically
reg reg
equivalent. In notation (10),
Aˆ “ A ¨¨¨A A , Bˆ “ B ¨¨¨B B .
i ri`t´1s ri`1s i i ri`t´1s ri`1s i
Let z be a natural number that satisfies both AˆzFmi “ Aˆz`1Fmi and BˆzFni “
i i i
Bˆz`1Fni for all i “ 1,...,t. By (3), the diagram
i
Fmi Aˆz //Fmi
ϕi ϕi
(cid:15)(cid:15) (cid:15)(cid:15)
Fni Bˆz // Fni
is commutative. Then ϕ ImAˆz “ ImBˆz for all i. Therefore, the restriction
i i i
ϕˆ : ImAˆz Ñ ImBˆz is a homeomorphism. The system of homeomorphisms
i i i
ϕˆ ,...,ϕˆ transforms A˜ to B˜, which are the regular parts of A and B by
1 t
Lemma 1(a).
Let us prove that r “ s, and, after a suitable renumbering, A and B are
i i
isomorphic for all i “ 1,...,r. Since all summands A and B with i ě 1 can
i i
be given by chains of basic vectors, it suffices to prove that n “ n1 for all i
ij ij
and j, where n1 is the number of singular summands B ,...,B in (8) given
ij 1 s
by chains of length j that end in the ith space Fni.
Due to (11), it suffices to prove that the numbers k are invariant with
ij
respect to topological equivalence.
In the same manner as k is constructed by A, we construct k1 by B.
ij ij
Let us fix i and j and prove that k “ k1 . Write
ij ij
A :“ A ¨¨¨A A , B :“ B ¨¨¨B B , q :“ ri`j `1s
ri`js ri`1s i ri`js ri`1s i
and consider the commutative diagram
Fmi A //Fmq
(12)
ϕi ϕq
(cid:15)(cid:15) (cid:15)(cid:15)
Fni B // Fnq
7
which is a fragment of (3). We have
k “ dimKerA “ m ´dimImA, k1 “ n ´dimImB.
ij i ij i
Because ϕ : Fmi Ñ Fni is a homeomorphism, m “ n (see [2, Corol-
i i i
lary 19.10] or [17, Section 11]). Since the diagram (12) is commutative,
ϕ pImAq “ ImB. Hence, the vector spaces ImA and ImB are homeomor-
q
phic, and so dimImA “ dimImB, which proves k “ k1 .
ij ij
(b) Each regular cycle A of the form (6) is isomorphic to the cycle A1 of
the form (9) since the diagram
Fm1 ii 1 //Fm2 1 //Fm3 1 //Fm4 1 // ... 1 //Fmt
At¨¨¨A2A1
1 A1 A2A1 A3A2A1 At´1¨¨¨A2A1 (13)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Fm1 ii A1 //Fm2 A2 // Fm3 A3 // Fm4 A4 // ... At´1 // Fmt
At
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 “
pϕ ,...,ϕ q of homeomorphisms; see (3). By (4),
1 t
ϕ A A ¨¨¨A “ B ϕ A ¨¨¨A
1 t t´1 1 t t t´1 1
“ B B ϕ A ¨¨¨A “ ¨¨¨ “ B B ¨¨¨B ϕ ,
t t´1 t´1 t´2 1 t t´1 1 1
and so the cycles Fm1ýA ¨¨¨A A and Fm1ýB ¨¨¨B B are topologically
t 2 1 t 2 1
equivalent via ϕ .
1
Conversely, let Fm1ýA ¨¨¨A A and Fm1ýB ¨¨¨B B be topologically
t 2 1 t 2 1
equivalent via some homeomorphism ϕ , and let A1 and B1 be constructed
1
by A and B as in (9). Then A1 and B1 are topologically equivalent via the
system of homeomorphisms ϕ “ pϕ ,ϕ ,...,ϕ q. Let ε and δ be systems of
1 1 1
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.
8
References
[1] J. Blanc, Conjugacy classes of affine automorphisms of Kn and linear
automorphisms of Pn in the Cremona groups, Manuscripta Math. 119
(2) (2006) 225–241.
[2] G.E. Bredon, Topology and Geometry, New York: Springer-Verlag,
1993.
[3] T.V. Budnitska, Classification of topological conjugate affine mappings,
Ukrainian Math. J. 61 (2009) 164–170.
[4] T. Budnitska, Topological classification of affine operators on unitary
and Euclidean spaces, Linear Algebra Appl. 434 (2011) 582–592.
[5] T. Budnitska, N. Budnitska, Classification of affine operators up to
biregular conjugacy, Linear Algebra Appl. 434 (2011) 1195–1199.
[6] S.E. Cappell, J.L. Shaneson, Linear algebra and topology, Bull. Amer.
Math. Soc., New Series, 1 (1979) 685–687.
[7] S.E. Cappell, J.L. Shaneson, Nonlinear similarity of matrices, Bull.
Amer. Math. Soc., New Series, 1 (1979) 899–902.
[8] S.E. Cappell, J.L. Shaneson, Non-linear similarity, Ann. of Math. 113
(2) (1981) 315–355.
[9] S.E. Cappell, J.L. Shaneson, Non-linear similarity and linear similarity
are equivariant below dimension 6, Contemp. Math. 231 (1999) 59–66.
[10] S.E. Cappell, J.L. Shaneson, M. Steinberger, J.E. West, Nonlinear sim-
ilarity begins in dimension six, Amer. J. Math. 111 (1989) 717–752.
[11] N.M. Dobrovol1skaya, V.A. Ponomarev, A pair of counter-operators, Us-
pehi Mat. Nauk. 20 (6) 80–86 (in Russian).
[12] W. Ephra¨mowitsch, Topologische Klassifikation affiner Abbildungen der
Ebene, Mat. Sb. 42 (1) (1935) 23–36.
[13] P. Gabriel, A.V. Roiter Representations of Finite-Dimensional Algebras,
Encyclopaedia of Math. Sci., V. 73 (Algebra VIII), Berlin: Springer-
Verlag, 1992.
9
[14] R.A. Horn, D.I. Merino, Contragredient equivalence: a canonical form
and some applications, Linear Algebra Appl. 214 (1995) 43–92.
[15] W.C. Hsiang, W. Pardon, When are topologically equivalent orthogonal
transformations linearly equivalent, Invent. Math. 68 (2) (1982) 275–
316.
[16] N.H. Kuiper, J.W. Robbin, Topological classification of linear endomor-
phisms, Invent. Math. 19 (2) (1973) 83–106.
[17] J. McCleary, A First Course in Topology: Continuity and Dimension,
Providence: Amer. Math. Soc., 2006.
[18] J. W.Robbin, Topologicalconjugacyandstructural stability fordiscrete
dynamical systems, Bull. Amer. Math. Soc. 78 (1972) 923–952.
[19] T. Rybalkina, Topological classification of pairs of counter linear maps,
Mat. Stud. 39 (1) (2013) 21–28 (in Ukrainian).
[20] T. Rybalkina, V.V. Sergeichuk, Topological classification of chains of
linear mappings, Linear Algebra Appl. 437 (2012) 860–869.
[21] V.V. Sergeichuk, Computation of canonical matrices for chains and cy-
cles of linear mappings, Linear Algebra Appl. 376 (2004) 235–263.
10