Topological similarity of nonsingular linear operators with root-of-unity eigenvalues: the missing case for the topological classification of matrix pencils

Futorny, Rybalkina, and Sergeichuk (2014) proved that the topological classification of matrix pencils — pairs of linear mappings classified up to homeomorphisms of the underlying vector spaces — reduces, via the regularizing (Kronecker-type) decomposition, to classifying the nonsingular regular part up to topological similarity, i.e. conjugacy by an arbitrary homeomorphism, and they identified the operator classification as an open problem solved only in special cases. Topological similarity of nonsingular operators is completely classified when no eigenvalue is a root of unity (Kuiper–Robbin over the reals; Budnitska over the complexes) and for finite-order operators over the reals, where Reidemeister torsion invariants determine the classification (Hambleton–Pedersen). The remaining open core — operators whose root-of-unity eigenvalues carry nontrivial Jordan structure or coexist with other spectral types, over both $\mathbb{R}$ and $\mathbb{C}$ — is unsettled, and resolving it is equivalent to completing the topological classification of matrix pencils.

Background

A matrix pencil is a one-parameter family of matrices $A+\lambda B$; two pencils are strictly equivalent when $S(A+\lambda B)R=A'+\lambda B'$ for nonsingular matrices $S,R$, which is the same as equivalence of the corresponding pairs of linear mappings $(A,B)$ and $(A',B')$ from $\mathbb{F}^n$ to $\mathbb{F}^m$. The classical Kronecker canonical theorem decomposes any pair into a regular part $(I_r,D)$ with $D$ a nonsingular $r\times r$ matrix, plus indecomposable singular blocks of four explicit types. Futorny–Rybalkina–Sergeichuk (2014) gave a constructive method for this regularizing decomposition and proved (their Theorem 1) that two pairs over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ are topologically equivalent — meaning there exist homeomorphisms $\varphi$ of $\mathbb{F}^n$ and $\psi$ of $\mathbb{F}^m$ with $A'\varphi=\psi A$ and $B'\varphi=\psi B$ — if and only if the singular summands of the two decompositions coincide up to permutation and the nonsingular matrices $D,D'$ are topologically similar. Thus the entire difficulty of classifying pencils under arbitrary topological coordinate changes concentrates in a single classical operator problem.

Two nonsingular operators $D,D':\mathbb{F}^r\to\mathbb{F}^r$ are topologically similar if some homeomorphism $h$ of $\mathbb{F}^r$ satisfies $h(Dx)=D'(h(x))$ for all $x$. Linear similarity always implies topological similarity, but the converse fails in a structured way. Kuiper and Robbin (1973) proved, for real matrices none of whose eigenvalues is a root of unity (a complex number $\zeta$ with $\zeta^m=1$ for some $m\ge 1$), a complete criterion: $D$ and $D'$ are topologically similar exactly when the dimensions and determinant signs of the contracting invariant subspaces (the spans of the generalized eigenspaces with $|\lambda|<1$) and of the expanding invariant subspaces ($|\lambda|>1$) match and the parts with eigenvalues of modulus $1$ are linearly similar. Budnitska (2011) extended the criterion to complex matrices without root-of-unity eigenvalues — there the unit-modulus parts are compared after realification (identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$), the condition being that the unit-modulus parts $A_1,B_1$ satisfy $A_1\oplus A_1$ similar to $B_1\oplus B_1$ — and produced canonical forms. Robbin (1972) organized the general question through the spectral splitting into a hyperbolic part (no eigenvalues of modulus 1) and a totally nonhyperbolic part (all eigenvalues of modulus 1): the hyperbolic contribution is fully captured by dimension-and-orientation invariants, and the whole residual obstruction lies in the unit-circle spectrum — precisely, in roots of unity.

When eigenvalues are roots of unity the operator has periodic structure. An operator of finite order ($D^m=I$) is the same thing as a real linear representation of the cyclic group of order $m$, and the classical question, studied from the 1930s onward, asks whether topologically equivalent representations must be linearly isomorphic. The answer is no, and nonlinear similarity — topological conjugacy without linear similarity — begins in dimension exactly six: Cappell–Shaneson (1981) constructed nonlinearly similar representations of the cyclic groups of order $4q\ge 8$ (in dimension $4q$), Cappell–Shaneson–Steinberger–West (1989) constructed six-dimensional examples for the cyclic groups of order $2^r$ with $r\ge 4$, and Cappell–Shaneson proved, in a 1981 preprint published in 1999, that no nonlinear similarity exists in dimension at most five for any finite group. Hambleton–Pedersen (2005, parts I and II) then completed the classification for all finite cyclic groups over $\mathbb{R}$: Reidemeister torsion — a combinatorial, homeomorphism-invariant of a based chain complex, computable in this setting as products $\prod(t^{a_i}-1)$ in the integral group ring $\mathbb{Z}[\mathbb{C}_m]$, evaluated in cyclotomic integer rings up to explicit indeterminacy — together with congruences on the representation weights determines topological equivalence, with a complete answer for cyclic $2$-power order groups. So the semisimple periodic case is settled over the reals.

What remains is the non-periodic root-of-unity case. A general nonsingular operator need not be of finite order: root-of-unity eigenvalues may carry nontrivial Jordan blocks (unipotent primary components, such as $I+N$ with $N$ nilpotent and nonzero), and they may coexist with contracting, expanding, and irrational-rotation eigenvalues. Robbin's Example 3.6 shows that unipotent Jordan structure is partially visible to topology: of the two unipotent real $4\times 4$ matrices with Jordan types $(2,2)$ and $(3,1)$ at the eigenvalue $1$, only one has a Hausdorff orbit space of the complement of the fixed-point set, so they are not topologically similar — but no complete invariant system or criterion is known for these configurations, over either field. Over $\mathbb{C}$, moreover, even the semisimple finite-order case is not covered by the settled results recorded above: the Hambleton–Pedersen classification is stated over $\mathbb{R}$, and Budnitska's criterion explicitly excludes root-of-unity eigenvalues. This is exactly the case left open by the 2014 reduction, and settling it is equivalent to the full topological classification of matrix pencils.

Problem Statement

Give a complete classification of nonsingular linear operators on finite-dimensional vector spaces over $\mathbb{R}$ (and, separately, over $\mathbb{C}$) up to topological similarity: necessary and sufficient conditions, in terms of effectively computable invariants of the two matrices, for the existence of a homeomorphism $h$ of $\mathbb{F}^r$ satisfying $h(Dx)=D'(h(x))$ for all $x\in\mathbb{F}^r$. The cases in which the spectrum contains no root of unity (Kuiper–Robbin for $\mathbb{R}$; Budnitska for $\mathbb{C}$) and the case of finite-order operators over $\mathbb{R}$ (Hambleton–Pedersen, via Reidemeister torsion) are already settled and any solution must reproduce them. The outstanding core concerns operators with root-of-unity eigenvalues beyond these settled cases: root-of-unity eigenvalues carrying nontrivial Jordan (unipotent) blocks, spectra mixing root-of-unity eigenvalues with contracting, expanding, or non-root unit-modulus eigenvalues, and — over $\mathbb{C}$ — the finite-order case, which the cited results settle only over $\mathbb{R}$. A solution must cover all of these configurations in arbitrary dimension and, by Theorem 1 of Futorny–Rybalkina–Sergeichuk (2014), would be exactly the missing ingredient for the topological classification of arbitrary matrix pencils (equivalently, pairs of linear mappings) over $\mathbb{R}$ and $\mathbb{C}$ up to homeomorphisms of the underlying spaces.

References

1. Vyacheslav Futorny, Tetiana Rybalkina, Vladimir V. Sergeichuk (2014), Regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings, Linear Algebra and its Applications 450:121–137, DOI 10.1016/j.laa.2014.03.002, https://doi.org/10.1016/j.laa.2014.03.002 (preprint arXiv:1403.2645, https://arxiv.org/abs/1403.2645)
2. N. H. Kuiper, J. W. Robbin (1973), Topological classification of linear endomorphisms, Inventiones Mathematicae 19(2):83–106, DOI 10.1007/BF01418922, https://doi.org/10.1007/BF01418922
3. J. W. Robbin (1972), Topological conjugacy and structural stability for discrete dynamical systems, Bulletin of the American Mathematical Society 78(6):923–953, DOI 10.1090/S0002-9904-1972-13058-1, https://doi.org/10.1090/S0002-9904-1972-13058-1
4. Sylvain E. Cappell, Julius L. Shaneson (1981), Non-linear similarity, Annals of Mathematics 113(2):315–355, DOI 10.2307/2006986, https://doi.org/10.2307/2006986
5. Sylvain E. Cappell, Julius L. Shaneson, Mark Steinberger, James E. West (1989), Nonlinear similarity begins in dimension six, American Journal of Mathematics 111(5):717–752, DOI 10.2307/2374878, https://doi.org/10.2307/2374878
6. Sylvain E. Cappell, Julius L. Shaneson (1999), Non-linear similarity and linear similarity are equivariant below dimension 6, in: Tel Aviv Topology Conference: Rothenberg Festschrift, Contemporary Mathematics 231, American Mathematical Society, Providence, RI, 59–66, DOI 10.1090/conm/231/03352, https://doi.org/10.1090/conm/231/03352
7. W.-c. Hsiang, William Pardon (1982), When are topologically equivalent orthogonal transformations linearly equivalent?, Inventiones Mathematicae 68(2):275–316, DOI 10.1007/BF01394060, https://doi.org/10.1007/BF01394060
8. Ian Hambleton, Erik K. Pedersen (2005), Topological equivalence of linear representations for cyclic groups: I, Annals of Mathematics 161(1):61–104, DOI 10.4007/annals.2005.161.61, https://doi.org/10.4007/annals.2005.161.61 (the preprint arXiv:math/0109036, https://arxiv.org/abs/math/0109036, combines Parts I and II)
9. Ian Hambleton, Erik K. Pedersen (2005), Topological equivalence of linear representations for cyclic groups: II, Forum Mathematicum 17(6):959–1010, DOI 10.1515/form.2005.17.6.959, https://doi.org/10.1515/form.2005.17.6.959
10. Tetiana Budnitska (2011), Topological classification of affine operators on unitary and Euclidean spaces, Linear Algebra and its Applications 434(2):582–592, DOI 10.1016/j.laa.2010.09.009, https://doi.org/10.1016/j.laa.2010.09.009 (preprint arXiv:1010.3380, https://arxiv.org/abs/1010.3380)
11. T. V. Rybalkina, V. V. Sergeichuk (2015), Topological classification of the oriented cycles of linear mappings, Ukrainian Mathematical Journal 66(10):1575–1581, DOI 10.1007/s11253-015-1032-1, https://doi.org/10.1007/s11253-015-1032-1 (preprint arXiv:1401.2550, https://arxiv.org/abs/1401.2550)
12. Carlos M. da Fonseca, Vyacheslav Futorny, Tetiana Rybalkina, Vladimir V. Sergeichuk (2017), Topological classification of systems of bilinear and sesquilinear forms, Linear Algebra and its Applications 515:1–5, DOI 10.1016/j.laa.2016.11.012, https://doi.org/10.1016/j.laa.2016.11.012 (preprint arXiv:1611.08716, https://arxiv.org/abs/1611.08716)
13. Carlos M. da Fonseca, Tetiana Rybalkina, Vladimir V. Sergeichuk (2016), Topological classification of sesquilinear forms: Reduction to the nonsingular case, Linear Algebra and its Applications 504:581–589, DOI 10.1016/j.laa.2016.04.022, https://doi.org/10.1016/j.laa.2016.04.022 (preprint arXiv:1604.05403, https://arxiv.org/abs/1604.05403)
14. Lvlin Luo (2021), Dynamical classification for complex matrices, Annals of Functional Analysis 12(2), article 20, DOI 10.1007/s43034-020-00106-5, https://doi.org/10.1007/s43034-020-00106-5