Topological conjugacy of periodic linear maps: the unresolved mixed-order cyclic cases of the nonlinear similarity classification

The 2008 survey 'Topological conjugacy classes of affine maps' (Budnytska, arXiv:0812.4921) recorded three open problems: extend the topological conjugacy classification of affine maps on $\mathbb{C}^n$ beyond $n\le 2$, determine the invariants distinguishing fixed-point-free affine maps for $n\ge 3$, and complete the topological classification of linear maps whose eigenvalues are roots of unity (the periodic case). The open problem is the unconditional, explicit classification for cyclic groups of mixed order $m=2^r q$ ($r\ge 2$, $q$ odd $>1$): computing the torsion subgroup of the topological representation ring $R_{\mathrm{Top}}(\mathbb{Z}_m)$ and the unstable (dimension-sensitive) similarity classification for these orders, where current results are conditional on hypotheses such as odd class numbers of cyclotomic fields.

Background

A map $f:\mathbb{F}^n\to\mathbb{F}^n$, $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, is affine if $f(x)=Ax+b$; the matrix $A$ is its linear part. Two such maps are topologically conjugate if some homeomorphism $h$ satisfies $h\circ f=g\circ h$; this is the coarsest natural equivalence for discrete dynamical systems, since conjugate maps have identical orbit structures. An affine map has a fixed point $p$ (with $f(p)=p$) exactly when the translation part can be removed: $f$ is topologically conjugate to its linear part $x\mapsto Ax$ whenever a fixed point exists. The source survey (Budnytska 2008) proved that for $n\le 2$ the fixed-point-free affine maps split into just two conjugacy classes distinguished by whether the linear part is singular, and posed the extension to $n\ge 3$. That extension is now closed: Budnitska (2011) proved that every fixed-point-free affine operator on $\mathbb{C}^n$ or $\mathbb{R}^n$, in any dimension, is topologically conjugate to exactly one canonical form $(I_k\oplus J_0)x+e_1$ (over $\mathbb{R}$, additionally with a single $[-1]$ block when the invertible part of $A$ has negative determinant), where $J_0$ is the nilpotent Jordan part of $A$; hence two such maps are conjugate exactly when their nilpotent parts are similar (and, over $\mathbb{R}$, the determinants of the invertible parts have the same sign). This also shows the two-class dichotomy genuinely fails from dimension 3 on, where the nilpotent Jordan type becomes an invariant. What remains open is the case the affine program inherits from the linear theory: affine maps with fixed points are conjugate precisely when their linear parts are, and the topological conjugacy classification of linear maps is still incomplete exactly for the periodic case. A linear map is periodic when it has finite order, that is, $A^m=I$ for some $m\ge 1$ (equivalently, $A$ is diagonalizable over $\mathbb{C}$ and all its eigenvalues are roots of unity). Kuiper and Robbin (1973) showed the general classification reduces to this case and conjectured that periodic linear maps are topologically conjugate if and only if linearly conjugate, proving it for periods $1,2,3,4,6$. Cappell and Shaneson (1979, 1981) disproved the conjecture — topologically conjugate periodic maps need not be linearly conjugate, a phenomenon called nonlinear similarity — and proved equivalence below dimension 6; Cappell–Shaneson–Steinberger–West (1989) showed nonlinear similarity first occurs in dimension 6. A periodic linear map of order $m$ is the same thing as a real representation of the finite cyclic group $\mathbb{Z}_m$, and conjugacy of two such maps is exactly $\mathbb{Z}_m$-equivariant homeomorphism of the representation spaces. Hsiang–Pardon (1982) and Madsen–Rothenberg (1982) proved that for $\mathbb{Z}_m$ with $m$ odd or twice odd, topological equivalence forces linear equivalence, and Cappell–Shaneson–Steinberger–Weinberger–West (1990) classified nonlinear similarities over $\mathbb{Z}_{2^r}$. The decisive later advance is due to Hambleton–Pedersen (2005), completing a program of de Rham: for a finite cyclic group $G$, nonlinear similarity is entirely determined by explicit congruences in the weights of the free summands (a representation is free if every non-identity group element fixes only the origin) together with the ratio of Reidemeister torsion invariants — a computable combinatorial invariant of equivariant complexes, valued in the Whitehead group $\mathrm{Wh}(\mathbb{Z}G)$, an algebraic $K$-theory quotient of $K_1$ — up to an explicitly described indeterminacy given by coboundary maps in Tate cohomology (the $2$-periodic cohomology theory of a finite group). Equivalently, in the topological representation ring $R_{\mathrm{Top}}(G)$ (the ordinary real representation ring modulo differences of topologically equivalent representations), the rank is known and the torsion subgroup encodes the stable similarity classification. Hambleton–Pedersen obtain a complete classification for cyclic groups of 2-power order; for $G=\mathbb{Z}_{4q}$ with $q$ odd their computation of the torsion subgroup is conditional on the fields $\mathbb{Q}(\zeta_d)$, $d\mid 4q$ ($\zeta_d$ a primitive $d$-th root of unity), having odd class number — true for $q\le 11$, false already for $q=29$ — and for general cyclic groups they determine the torsion subgroup only up to explicit criteria that are evaluated in the 2-power case and, conditionally, for $\mathbb{Z}_{4q}$.

Problem Statement

Two linear maps $f(x)=Ax$ and $g(x)=Cx$ on $\mathbb{R}^n$ (or on $\mathbb{C}^n$, viewed through its underlying real structure) are topologically conjugate if some homeomorphism $h$ satisfies $h\circ f=g\circ h$. When the eigenvalues of $A$ and $C$ avoid the roots of unity this conjugacy relation is completely classified (Kuiper–Robbin), and after Budnitska (2011) the same holds for all affine maps except insofar as their linear parts fall into the periodic case — a linear map is periodic when it has finite order, that is, $A^m=I$ for some $m\ge 1$. A periodic linear map of order $m$ is a real representation of the cyclic group $\mathbb{Z}_m$, and topological conjugacy of two such maps is the same as topological similarity of the associated representations, that is, a $\mathbb{Z}_m$-equivariant homeomorphism of the representation spaces; when the two representations are not linearly conjugate, such a topological similarity is called a nonlinear similarity. The remaining open problem is:

Determine, unconditionally and explicitly, the topological conjugacy classification of periodic linear maps, equivalently the nonlinear similarity classification of real representations of $\mathbb{Z}_m$, for the cyclic groups of mixed order $m=2^r q$ with $r\ge 2$ and $q$ odd, $q>1$ (the unconditional results for, e.g., $\mathbb{Z}_{4q}$ with $q\le 11$ settle the stable torsion subgroup and exclude 6-dimensional nonlinear similarities, but do not give the full unstable classification). The classification comprises both of the following:

1. Compute the torsion subgroup of the topological representation ring $R_{\mathrm{Top}}(\mathbb{Z}_m)$ for these orders — equivalently, determine exactly which differences of free representations become equal under stable topological similarity — removing the number-theoretic hypotheses (such as odd class numbers of the cyclotomic fields $\mathbb{Q}(\zeta_d)$, $d\mid m$) under which the currently known results for $\mathbb{Z}_{4q}$ are stated.

2. Give the corresponding unstable (dimension-sensitive) classification: an effective criterion deciding whether two given periodic linear maps of such order $m$ are topologically conjugate, including the minimal-dimension questions, in particular whether 6-dimensional nonlinear similarities exist for $\mathbb{Z}_{4q}$ with $q$ odd (known not to exist when the relevant class numbers are odd; open for orders such as those involving $q=29$).

References

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