Dynamical obstructions to classification by (co)homology and other TSI-group invariants

One of the leading questions in many mathematical research programs is whether a certain classification problem admits a “satisfactory” solution. What constitutes a satisfactory solution depends of course on the context, and it is often subject to change when the original goals are deemed hopeless. Invariant descriptive set theory provides a formal framework for measuring the complexity of classification problems and for showing which types of invariants are inadequate for complete classification.

Hjorth’s turbulence theory is one of the biggest accomplishments in this direction as it provides obstructions to classification by countable structures, i.e., for classification using only isomorphism types of countable structures as invariants. Historically, this type of classification played important role in ergodic theory since von Neumann’s seminal paper [vN32], where he shows that ergodic measure-preserving transformations of discrete spectrum are completely classified up to isomorphism by their (countable) spectrum. This led to the hope that, by adding further structure on these countable invariants, one could classify all ergodic measure-preserving transformations via countable structures. However, 70 years after the publication of [vN32], the newly developed theory of turbulence [Hjo00] was used to establish that this hope was too optimistic; see [Hjo01, FW04].

Another very common type of classification that one encounters in mathematical practice is classification by (co)homological invariants, i.e., classification using elements of homology or cohomology groups of some appropriate chain complex. For example:

1. (1)

   there is an assignment $p\mapsto c(p)$, from Hermitian line bundles $p\colon E\to B$ over a locally compact metrizable space $B$, to the Čech cohomology group $\mathrm{H}^{2}(B)$ of $B$, so that $p$ and $p^{\prime}$ are isomorphic over $B$ iff $c(p)=c(p^{\prime})$; see [RW98].

2. (2)

   there is an assignment $\mathcal{A}\mapsto c(\mathcal{A})$, from continuous-trace $C^{*}$-algebras $\mathcal{A}$ with locally compact metrizable spectrum $S$, to the Čech cohomology group $\mathrm{H}^{3}(S)$ of $S$, so that $\mathcal{A}$ and $\mathcal{A}^{\prime}$ are Morita equivalent over $S$, iff $c(\mathcal{A})=c(\mathcal{A}^{\prime})$; see [Bla09].

3. (3)

   there is an assignment $e\mapsto c(e)$, from group extensions $e\colon E\to\mathbb{Q}_{2}$ of the dyadic rationals $\mathbb{Q}_{2}$ by $\mathbb{Z}$, to the Steenrod homology group $\mathrm{H}_{0}(\Sigma)$ of the character $\Sigma$ of $\mathbb{Q}_{2}$, so that $e$ and $e^{\prime}$ are isomorphic iff $c(e)=c(e^{\prime})$; see [EM42].

Up until recently, classification by (co)homological invariants had not been considered from the perspective of invariant descriptive set theory. However, as a consequence of the results in [BLP19], classification by the above (co)homological invariants forms a well defined complexity class in the standard setup of invariant descriptive set theory. This complexity class is entirely contained within the class of all classification problems which are classifiable by abelian group actions; see Problem 1. As a consequence, Corollary 1.4 below provides a new anti-classification criterion, in the spirit of Hjorth’s turbulence theory, which can be used as an obstruction to classification by (co)homological invariants. To illustrate its use, we will apply it to show that the coordinate free versions of Hermitian line bundle isomorphism and of Morita equivalence between continuous-trace $C^{*}$-algebras cannot be classified by (co)homological invariants as in the examples (1) and (2) above.

The formal framework that is often used for measuring the complexity of classification problems is the Borel reduction hierarchy. Formally, a classification problem is a pair $(X,E)$, where $X$ is a Polish space and $E$ is an analytic equivalence relation. A classification problem $(X,E)$ is considered to be of “less or equal complexity” to the classification problem $(Y,F)$, if there is a Borel reduction from $E$ to $F$, i.e., a Borel map $f\colon X\to Y$, so that for all $x,x^{\prime}\in X$ we have:

At the lower end of this complexity hierarchy we have—in increasing complexity—the classification problems which are: concretely classifiable; essentially countable; and classifiable by countable structures. A classification problem $(X,E)$ is concrete classifiable if $E$ Borel reduces to the equality relation of some Polish space. It is essentially countable if it Borel reduces to a Borel equivalence relation $F$ which has equivalence classes of countable size. We finally say that $(X,E)$ is classifiable by countable structures if $(X,E)$ Borel reduces to the problem $(X_{\mathcal{L}},\simeq_{\mathrm{iso}})$, of classifying, up to isomorphism, all countable $\mathcal{L}$-structures (graphs, groups, rings, etc.) of some fixed language $\mathcal{L}$.

In order to show that two classification problems differ in complexity, it is imperative to have a basic obstruction theory for Borel reductions. These obstructions often come from dynamics and they directly apply to classification problems of the form $(X,E^{G}_{X})$, where $E^{G}_{X}$ is the orbit equivalence relation of the continuous action of a Polish group $G$ on a Polish space $X$. A classical dynamical obstruction to concrete classification is generic ergodicity; see [Gao09]. Similarly, Hjorth’s turbulence theory [Hjo00] provides obstructions to classifiability by countable structures. Finally, storminess [Hjo05], as well as local approximability [KMPZ20], are both dynamical obstruction to being essentially countable.

These dynamical obstructons can be seen to answer the following general problem that was considered in [LP18]: if $\mathcal{C}$ is a class of Polish groups, then we say that $(X,E)$ is classifiable by $\mathcal{C}$-group actions if it is Borel reducible to an orbit equivalence relation $(Y,E^{H}_{Y})$, where $H$ is a group from $\mathcal{C}$.

Indeed, generic ergodicity provides an answer for $\mathcal{C}=\{\text{compact Polish groups}\}$; turbulence for $\mathcal{C}=\{\text{non-Archimedean Polish groups}\}$; while storminess and local-approximability for the class $\mathcal{C}=\{\text{locally-compact Polish groups}\}$. More recently, an answer to this problem for the case where $\mathcal{C}$ is the class of all Polish CLI groups has been given [LP18]. Recall that a Polish group is CLI if it admits a complete and left-invariant metric.

The main goal of this paper is to provide an answer to the above problem for the case where $\mathcal{C}$ is the class of all Polish groups which admit a two side invariant (TSI) metric. Since every abelian group is TSI the obstructions we introduce can be used for showing when a classification problem is not classifiable by (co)homological invariants.

A Polish space $X$ is a separable, completely metrizable topological space. A Polish group $G$ is a topological group whose topology is Polish. Let $d$ be a metric on $G$ that is compatible with the topology. We say that $d$ is left invariant if $d(gh,gh^{\prime})=d(h,h^{\prime})$, for all $g,h,h^{\prime}\in G$ and right invariant if $d(hg,h^{\prime}g)=d(h,h^{\prime})$, for all $g,h,h^{\prime}\in G$. We say that $d$ is two side invariant, if it is both left and right invariant. We say that a Polish group is TSI, if it admits a two sided invariant metric which is compatible with the topology. Such groups are often called balanced since, by a theorem of Klee, they are precisely the Polish groups which admit a neighborhood basis of the identity consisting of conjugation-invariant open sets. We say that $G$ is non-Archimedean, if it admits a basis of open neighborhoods of the identity consisting of open subgroups. A Polish $G$-space is a Polish space $X$ together with a continuous left action of a Polish group $G$ on $X$. If $x\in X$, we write $[x]$ to denote the orbit $Gx$ of $x$. We denote by $E^{G}_{X}$ the associated orbit equivalence relation:

It is clear that $x\leftrightsquigarrow y$ implies that $\overline{Gx}=\overline{Gy}$. It is also clear that this is a symmetric relation that is invariant under the action of $G$; that is, if $g,h\in G$, we have that $x\leftrightsquigarrow y$ if and only if $gx\leftrightsquigarrow hy$. As a consequence we can write $[x]\leftrightsquigarrow[y]$ whenever $x\leftrightsquigarrow y$, without ambiguity.

We say that $(C/G,\leftrightsquigarrow)$ is connected, if for every $x,y\in C$ there is a path in $C/G$ from $[x]$ to $[y]$; that is, a sequence $x_{0},\ldots,x_{n-1}\in C$ so that $x=x_{0},x_{n-1}=y$ and $x_{i-1}\leftrightsquigarrow x_{i}$, for all $0<i<n$. We say that $(X/G,\leftrightsquigarrow)$ is generically semi-connected, if for every $G$-invariant comeager $C\subseteq X$, there is a comeager $D\subseteq C$ so that for every $x,y\in D$ there is a path between $[x]$ and $[y]$ in $(C/G,\leftrightsquigarrow)$. We say that a Polish $G$-space $X$ is generically unbalanced, if it has meager orbits and $(X/G,\leftrightsquigarrow)$ is generically semi-connected.

Let $(X,E),(Y,F)$ be two classification problems. A Baire-measurable homomorphism from $E$ to $F$ is a a Baire-measurable map $f:X\rightarrow Y$ so that $xEx^{\prime}\implies f(x)Ff(x^{\prime})$. It is a Baire-measurable reduction, if we additionally have $f(x)Ff(x^{\prime})\implies xEx^{\prime}$. We say that $(X,E)$ is generically $F$-ergodic, if for every Baire-measurable homomorphism from $E$ to $F$ there is a comeager subset $C\subseteq X$ so that $f(x)F(x^{\prime})$ for all $x,x^{\prime}\in C$. The following theorem and its corollary are the main results of this paper.

We now turn to applications. In [CC], a new family of jump operators $E\mapsto E^{[\Gamma]}$ was introduced which are similar to the Friedman–Stanley jump $E\mapsto E^{+}$: for every countable group $\Gamma$, the $\Gamma$-jump of the classification problem $(X,E)$ is the classification problem

In the same paper they showed that the $\mathbb{Z}$-jump $E^{[\mathbb{Z}]}_{0}$ of $E_{0}$ is generically ergodic with respect to the countable product $E^{\omega}_{0}$ of $E_{0}$. In [All20], the stronger result was shown that, in fact, $E^{[\mathbb{Z}]}_{0}$ is generically $E^{H}_{Y}$-ergodic, whenever $H$ is both a non-Archimedean and TSI Polish group. As a consequence of Theorem 1.3, we now have that $E^{[\mathbb{Z}]}_{0}$ is generically $E^{H}_{Y}$-ergodic for every TSI Polish group $H$ and therefore $E^{[\mathbb{Z}]}_{0}$ is not classifiable by any TSI-group action. In fact, in Section 2 we define a $P$-jump operator $E\mapsto E^{[P]}$, for every Polish permutation group $P$, which is a common generalization of the Friedman–Stanley jump and the jump operators defined in [CC]. It turns out that $P$-jumps are particularly natural in the context of the generalized Bernoulli shifts from [KMPZ20]: if $E$ is the orbit equivalence relation of the generalized Bernoulli shift of the Polish permutation group $Q$, then $E^{[P]}$ is the orbit equivalence relation of the generalized Bernoulli shift of $(P\text{ Wr }Q)$. The anti-classification result for $E^{[\mathbb{Z}]}_{0}$ is a particular instance of the next corollary. Here, $(P\text{ Wr }G)$ is just the Wreath product of $P$ and $G$; see Section 2.

Corollary 1.6 provides many examples of orbit equivalence relations of CLI groups which are not classifiable by TSI-group actions. However, all such examples are classifiable by countable structures. The next corollary shows that the complexity class within the CLI region but outside of the TSI and the non-Archimedean region in Figure 1 is non-empty:

We finally illustrate how our results apply to natural classification problems from topology and operator algebras. For any locally compact metrizable space $T$, consider the problem $(\mathrm{CTr}^{*}(T),\equiv_{\mathcal{M}})$, of classifying all separable continuous-trace $C^{*}$-algebras with spectrum $T$ up to Morita equivalence; and the problem $(\mathrm{Bun}_{\mathbb{C}}(T),\simeq_{\mathrm{iso}})$, of classifying all Hermitian line bundles over $T$ up to isomorphism. In Section 6 we show that both problems are in general not classifiable by actions of TSI-groups, even when $T$ is a CW-complex. In contrast, recall that the base-preserving versions $\equiv_{\mathcal{M}}^{T}$, and $\simeq_{\mathrm{iso}}^{T}$, of the above problems are always classifiable by TSI—in fact by abelian—group actions; [BLP19].

Theorem 1.3, which is the main result of this paper, is proved in Section 5. The necessary background is developed independently in Section 3 and Section 4. In Section 2 we assume Theorem 1.3 and provide “in vitro” applications, such as Theorem 1.5, Corollary 1.6, and Corollary 1.7. In Section 6 we discuss applications in topology and operator algebras. Finally, in Section 7 we informally announce some extensions of this work beyond the TSI dividing line which is currently work in progress.