Dynamical obstructions to classification by (co)homology and other TSI-group invariants
aa r X i v : . [ m a t h . L O ] A p r DYNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONSOF TSI GROUPS
SHAUN ALLISON AND ARISTOTELIS PANAGIOTOPOULOS
Abstract.
We introduce a dynamical condition for Polish G -spaces, in the spirit ofHjorth’s turbulence theory, which implies non-classifiability by actions of Polish TSI-groups. These are the groups which admit a metric that is invariant from both sides.We show that the Wreath product of any two non-compact subgroups of S ∞ admitsan action whose orbit equivalence relation is generically ergodic against actions of TSI-groups, and deduce that there is an orbit equivalence relation of a CLI group which is notclassifiable by TSI group actions. Finally, we show that Morita equivalence of continuous-trace C ∗ -algebras, as well as isomorphism of Hermitian line bundles, are not classifiableby TSI-group actions. Introduction
Background.
Significant mathematical activity revolves around classification prob-lems. However, not every classification problem has a satisfactory solution and some clas-sification problems are more complicated than others. A formal framework that is oftenused for measuring the complexity of classification problems is the Borel reduction hier-archy. Formally, a classification problem is a pair (
X, E ), where X is a Polish spaceand E is an analytic equivalence relation. A classification problem ( X, E ) is considered tobe of “less or equal complexity” to the classification problem (
Y, F ), if there is a
Borelreduction from E to F , i.e., a Borel map f : X → Y , so that for all x, x ′ ∈ X we have: xEx ′ ⇐⇒ f ( x ) F f ( x ′ ) . For example, consider the problem of classifying all Hermitian line bundles over a fixedlocally compact base space T , up to isomorphism; or the collection of all continuous-trace C ∗ -algebras of fixed spectrum T , up to Morita equivalence. Both problems are classificationproblems in the formal sense above. We can therefore attempt to measure their “intrinsiccomplexity” by finding their location within the Borel reduction complexity hierarchy. At Mathematics Subject Classification.
Primary 54H05, 37B02, 54H11; Secondary 46L35, 55R15.
Key words and phrases.
Polish group, invariant metric, generically ergodic, turbulence, TSI, CLI, Borelreduction, continuous-trace C ∗ -algebra, Morita equivalence, Hermitian line bundle.We would like to thank A. Shani, M. Lupini, J. Bergfalk, and A.S. Kechris for all the useful and inspiringdiscussions. We would also like to thank S. Coskey and J.D. Clemens for sharing their early draft [CC]with us. Finally, we want to acknowledge the hospitality and financial support of the California Instituteof Technology during the visit of S.A. in the winter of 2020. the lower end of this complexity hierarchy we have—in increasing complexity—the classifi-cation problems which are: concretely classifiable; essentially countable; and classifiable bycountable structures. A classification problem ( X, E ) is concrete classifiable if E Borelreduces to the equality relation of some Polish space. It is essentially countable if itBorel reduces to a Borel equivalence relation F which have equivalence classes of countablesize. We finally say that ( X, E ) is classifiable by countable structures if (
X, E ) Borelreduces to the problem ( X L , ≃ iso ), of classifying all countable L -structures (graphs, groups,rings, etc.) of some fixed language L , up to isomorphism.In order to show that two classification problems have different complexity, it is im-perative to have a basic obstruction theory for Borel reductions. These obstructions oftencome from dynamics and they directly apply to classification problems of the form ( X, E GX ),where E GX is the orbit equivalence relation of the continuous action of a Polish group G on aPolish X . A classical dynamical obstruction to concrete classification is generic ergodicity ;see [Gao09]. Similarly, Hjorth’s turbulence theory [Hjo00] provides obstructions to classifi-ability by countable structures. Finally, storminess [Hjo05], as well as local approximability [KMPZ19], are both dynamical obstruction to being essentially countable.These dynamical obstructons can be seen to answer the following general problem thatwas considered in [LP18]: if C is a class of Polish groups, then we say that ( X, E ) is classifiable by C -group actions if it is Borel reducible to an orbit equivalence relation( Y, E HY ), where H is a group from C . Problem 1.
Given a class C of Polish groups, which dynamical conditions on a Polish G -space X ensure that ( X, E GX ) is not classifiable by C -group actions? Indeed, generic ergodicity provides an answer for C = { compact Polish groups } ; turbu-lence for C = { non-Archimedean Polish groups } ; while storminess and local-approximabilityfor the class C = { locally-compact Polish groups } . More recently, an answer to this prob-lem for the case where C is the class of all Polish CLI groups has been given [LP18]. Recallthat a Polish group is CLI if it admits a complete and left-invariant metric.Polish CLITSInon-Archimedeanlocally compactcompact
Figure 1.
Classification by C -group actions for various group classes C . YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 3
In this paper, we develop further the “quotient structuring” techniques developed in[LP18] and provide an answer to the above problem for the case where C is the class ofall Polish groups which admit a two side invariant ( TSI ) metric. Turning to applications,in [All] it was shown that the Z -jump E [ Z ]0 of E (see Section 2) is generically ergodic forany orbit equivalence relation ( Y, E HY ), whenever H is both non-Archimedean and TSI. Weshow here that it is, in fact, generically ergodic for E H Y , even when H is just TSI. Thisshows that the CLI complexity class in Figure 1 is strictly bigger than the TSI complexityclass. We then establish that Morita equivalence between continuous trace C ∗ -algebras,as well as isomorphism between Hermitian line bundles is, in general, not classifiable byTSI-group actions.1.2. Definitions and main results. A Polish space X is a separable, completelymetrizable topological space. A Polish group G is a topological group whose topol-ogy is Polish. Let d be a metric on G that is compatible with the topology. We saythat d is left invariant if d ( gh, gh ′ ) = d ( h, h ′ ), for all g, h, h ′ ∈ G and right invariant if d ( h, h ′ g ) = d ( h, h ′ ), for all g, h, h ′ ∈ G . We say that d is two side invariant , if itis both left and right invariant. We say that a Polish group is TSI , if it admits a twosided 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 admita neighborhood basis of the identity consisting of conjugation invariant open sets. We saythat G is non-Archimedean , if it admits an basis of open neighborhoods of the identityconsisting of open subgroups. A Polish G -space is a Polish space X together with acontinuous left action of a Polish group G on X . If x ∈ X , we write [ x ] to denote the orbit Gx of x . We denote by E GX the associated orbit equivalence relation : xE GX y ⇐⇒ [ x ] = [ y ] . Definition 1.1.
Let X be a Polish G -space and let x, y ∈ X . We write x ! y , if forevery open neighborhood V of the identity of G and every open neighborhood U ⊆ X of x or y , there exist g x , g y ∈ G with g x x ∈ U and g y y ∈ U , so that:( g y y ) ∈ V ( g x x ) and ( g x x ) ∈ V ( g y y ) . It is clear that this is a symmetric relation that is invariant under the action of G ; thatis, if g, h ∈ G , we have that x ! y if and only if gx ! hy . As a consequence we canunambiguously write [ x ] ! [ y ], whenever x ! y . Definition 1.2.
Let X be a Polish G -space and let C ⊆ X be a G -invariant set. The unbalanced graph associated to the action of G on C is the graph ( C/G, ! ), where C/G := { [ x ] | x ∈ C } and [ x ] ! [ y ] if and only if x ! y .We say that ( C/G, ! ) is connected , if for every x, y ∈ C there is a path in C/G from [ x ] to [ y ]; that is, a sequence x , . . . , x n − ∈ C so that x = x , x n − = y and x i − ! x i , forall i < n . We say that ( X/G, ! ) is generically semi-connected , if for every comeager C ⊆ X , there is a comeager D ⊆ C so that for every x, y ∈ D there is a path between [ x ] SHAUN ALLISON AND ARISTOTELIS PANAGIOTOPOULOS and [ y ] in ( C/G, ! ). We say that a Polish G -space X is generically unbalanced , if ithas meager orbits and ( X/G, ! ) is generically semi-connected.Let ( X, E ) , ( Y, F ) be two classification problems. A
Baire-measurable homomor-phism from E to F is a a Baire-measurable map from X to Y so that xEx ′ = ⇒ f ( x ) F f ( x ′ ). It is a Baire-measurable reduction , if we additionally have f ( x ) F f ( x ′ ) = ⇒ xEx ′ . We say that ( X, E ) is generically ergodic , if for every Baire-measurable homomor-phism from E to F there is a comeager subset C ⊆ X so that f ( x ) F ( x ′ ) for all x, x ′ ∈ C .The following is theorem and its corrollary are the main results of this paper. Theorem 1.3.
Let X be a Polish G -space and let Y be a Polish H -space, where H is TSI.If ( X/G, ! ) is generically semi-connected, then E GX is generically E HY -ergodic. Corollary 1.4 (Obstruction to classification by TSI) . If the Polish G -space X is genericallyunbalanced, then the orbit equivalence relation E GX is not classifiable by TSI-group actions. We now turn to applications. In [CC], a new family of jump operators E E [Γ] wasintroduced which are similar to the Friedman–Stanley jump E E + : for every countablegroup Γ, the Γ -jump of the classification problem ( X, E ) is the classification problem( X Γ , E [Γ] ) , with xE [Γ] x ′ ⇐⇒ ( ∃ γ ∈ Γ) ( ∀ α ∈ Γ) x ( γ − α ) E x ′ ( α ) . In the same paper they showed that the Z -jump E [ Z ]0 of E is generically ergodic withrespect to the countable product E ω of E . In [All], this was strengthened to the fact that E [ Z ]0 is generically E HY -ergodic, whenever H is both a non-Archimedean and TSI Polishgroup. As a consequence of Theorem 1.3, we have that E [ Z ]0 is generically E HY -ergodic forevery TSI Polish group H and therefore E [ Z ]0 is not classifiable by any TSI-group action. Infact, in Section 2 we define a P -jump operator E E [ P ] , for every Polish permutationgroup P ≤ S ∞ , which is a common generalization of the Friedman–Stanley jump and thejump operators defined in [CC]. It turns out that P -jumps are particularly natural in thecontext of the generalized Bernoulli shifts from [KMPZ19]: if E is the orbit equivalencerelation of the generalized Bernoulli shift of the Polish permutation group Q , then E [ P ] isthe orbit equivalence relation of the generalized Bernoulli shift of ( P Wr N Q ). The anti-classification result for E [ Z ]0 is a particular instance of the next corollary. Here, ( P Wr N G )is just the Wreath product of P and G ; see Section 2. Theorem 1.5.
Let X be a Polish G -space which has a dense orbit, and let P be Polishgroup of permutations of a countable set N . If all P -orbits of N are infinite, then theunbalanced graph of the Polish ( P Wr N G ) -space X N is generically semi-connected. Corollary 1.6. If P, Q ≤ S ∞ are both non-compact Polish permutation groups then theBernoulli shift of ( P Wr N Q ) has a generically unbalanced closed subshift. Corollary 1.6 provides many examples of orbit equivalence relations of CLI groups whichare not classifiable by TSI-group actions. However, all such examples are classifiable bycountable structures. The next corollary shows that the complexity class within the CLIregion but outside of the TSI and the non-Archimedean region in Figure 1 is non-empty:
YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 5
Corollary 1.7.
There is a Polish G -space X of a CLI Polish group G which is turbulentand generically unbalanced. We finally illustrate how our results apply to natural classification problems from topol-ogy and operator algebras. For any locally compact metrizable space T , consider the prob-lem (CTr ∗ ( T ) , ≡ M ), of classifying all separable continuous-trace C ∗ -algebras withspectrum T up to Morita equivalence ; and the problem (Bun C ( T ) , ≃ iso ), of classifyingall Hermitian line bundles over T up to isomorphism . In Section 6 we show thatboth problems are, in general, not classifiable by actions of TSI-groups, even when T is aCW-complex. In contrast, recall that the “ base-preserving versions ≡ T M , and ≃ T iso , of theabove problems are always classifiable by TSI—in fact by abelian—group actions; [BLA19].1.3. Structure of the paper.
Theorem 1.3, which is the main result of this paper, isproved in Section 5. The necessary background is developed independently in Section 3and 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 applicationsin topology and operator algebras. Finally, in Section 7 we informally announce someextensions of this work, beyond the TSI dividing line, which is currently in work progress.2.
Wreath products, P -jumps, and Bernoulli shifts In this section, for every Polish permutation group P of a countable set we introducea P -jump operator for G -spaces and equivalence relations, in the spirit of [CC]. We thendraw some connections with the theory of generalized Bernoulli shifts from [KMPZ19]. Wealso prove Theorem 1.5, and derive Corollary 1.6 and Corollary 1.7.Let N be a countable set. We denote by S ( N ) the group of all permutations of N . Thisis a Polish group, when endowed with the pointwise convergence topology. By a Polishgroup of permutations of N we mean any closed subgroup P of S ( N ). Since we areconsidering left actions, the pertinent action P × N → N is given by( p, n ) pn := p − ( n ) , for any map p : N → N in P. Let P be as above and let G be an arbitrary Polish group. The group G N := { g : N → G } , where ( g g )( n ) := g ( n ) g ( n ) , is Polish and there is a natural P -action ϕ : P × G N → G N on G N by automorphisms: ϕ ( p, g ) = g p , where g p ( n ) := g ( p − · n ) = g ( p ( n )) . The
Wreath product ( P Wr N G ) of P and G , or simply ( P Wr G ), is the group P Wr G := P ⋊ ϕ G N . Concretely, elements of P Wr G are all pairs ( p, g ), where p ∈ P , g ∈ G N , and( p , g ) · ( p , g ) := ( p p , ϕ ( p − , g ) g ) , where (cid:0) ϕ ( p − , g ) g (cid:1) ( n ) = g ( p · n ) g ( n ) . If X is a Polish G -space, then the P -jump of G y X is the P Wr G -space X N :( p, g ) · x := x g ,p , where x g ,p ( n ) := g p ( n ) · x p ( n ) = g ( p ( n )) · x ( p ( n )) . SHAUN ALLISON AND ARISTOTELIS PANAGIOTOPOULOS
Similarly, let ( X N , E [ P ] ) be the P -jump of a classification problem ( X, E ), where: x E [ P ] x ′ ⇐⇒ ( ∃ p ∈ P ) ( ∀ n ∈ N ) x ( p ( n )) E x ′ ( n )Notice that if P = S ( N ), then this is simply the Friedman–Stanley jump E E + , and if P is the left regular representation of a countable group Γ as a subgroup of S (Γ), then thisis the Γ-jump E E [Γ] , introduced in [CC]. Clearly, the orbit equivalence relation of the P -jump of a G -space X is the P -jump of the orbit equivalence relation of the same space.We may now proceed to the proof of Theorem 1.5. Proof of Theorem 1.5.
Let X be a Polish G -space which has a dense orbit and let P ≤ S ( N )be a Polish permutation group on a countable set N with infinite orbits.Fix any comeager set C ⊆ X N . For any fixed n ∈ N consider “column” space X { n } ofall maps from the singleton { n } to X . This is naturally isomorphic to the Polish G -space X . By intersecting C with the appropriate comeager set we may assume without loss ofgenerality that for all x ∈ C we have that:(1) for all n ∈ N, the orbit of x ( n ) is dense in X { n } . Here we use that { x ∈ X | [ x ] is dense } is a G δ subset of X , and since by assumption itcontains a dense orbit, it is comeager.Let E be the collection of all bijective maps from N to the space N × { , } , of twodisjoint copies of N . Then E is a Polish space with the pointwise convergence topology.Since every P -orbit of N is infinite, by the Neuman’s lemma [Neu76, Lemma 2.3] we havethat for every finite A, B ⊆ N , there is p ∈ P so that ( p · A ) ∩ B = ∅ . As a consequence,for the generic e ∈ E and for every i ∈ { , } :(2) if A ⊆ N is finite, then there is p ∈ P so that e ( p · n ) = ( p · n, i ) , for all n ∈ A. Fix some e ∈ E satisfying the above and consider the map ϕ : X N × X N → X N , where ϕ ( x , x ) is the function x : N → X , with x ( n ) = x ( m ), if e ( n ) = ( m, x ( n ) = x ( m ), if e ( n ) = ( m, Claim 2.1.
The map ϕ : X N × X N → X N is a homeomorphism of topological spaces.Proof of Claim. Since e is bijective, every x ∈ X N pulls back to some ( x , x ) via ϕ andevery ( x , x ) ∈ X × X pushes forward to some x ∈ X , i.e., ϕ is a bijection. Continuity of ϕ and ϕ − is straightforward. (cid:3) Consider the comeager subset C ϕ := ϕ − ( C ) T ( C × C ) of X N × X N . By Kuratowski-Ulam, there is some z ∈ C so that D := { y ∈ X | ( z , y ) ∈ C ϕ } is a comeager subet of X .We are now done with the proof, since by the next claim, for every x , y ∈ D we have thefollowing path between x and y in C : x ! ϕ ( x , z ) ! z ! ϕ ( z , y ) ! y . Claim 2.2.
For all x , x ∈ C we have that ϕ ( x , x ) ! x and ϕ ( x , x ) ! x . YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 7
Proof of Claim.
It suffices to prove that ϕ ( x , x ) ! x since the other case is symmetric.Set x = x and y = ϕ ( x , x ). Let V ⊆ ( P Wr G ) be an open neighborhood of the identityand let U ⊆ X be any non-empty open set. By shrinking both sets if neccessary, we assumethat there is a finite set A ⊆ N , a map u : A → X , an open neighborhood W ⊆ G of theidentity of G , and some ε >
0, so that V = { ( p, g ) ∈ ( P Wr G ) | g ( n ) ∈ W, for all n ∈ A } , and U = { x ∈ X N | d (cid:0) x ( n ) , u ( n ) (cid:1) < ε, for all n ∈ A } , where d is any metric on X that is compatible with the topology.By (2) there is p ∈ P so that (cid:0) ( p, G ) · x (cid:1) ( n ) = (cid:0) ( p, G ) · y (cid:1) ( n ), for all n ∈ A . By (1)we may set g x = g y := ( p, g ) for some g ∈ G N so that in addition to(3) (cid:0) ( p, g ) · x (cid:1) ( n ) = (cid:0) ( p, g ) · y (cid:1) ( n ) , for all n ∈ A, we also have g x · x , g y · y ∈ U . But V contains the following subgroup of ( G Wr P ): { (1 P , h ) | h ( n ) = 1 G , for all n ∈ A } . By (1) and (3), it now follows that ( g y y ) ∈ V · ( g x · x ) and ( g x x ) ∈ V · ( g y · y ). (cid:3)(cid:3) Let P be a Polish permutation group of a countable set N . The generalized Bernoullishift of P is the Polish P -space R N where ( p, x ) x p with x p ( n ) = x ( p − n ) = x ( p ( n )),for every n ∈ N . This indeed generalizes the classical Bernoulli shift Γ y R Γ for countabledisrete groups, where ( g, x ) g · x with ( g · x )( γ ) = x ( g − γ ). In [KMPZ19] it was shownthat the Borel reduction complexity of the orbit equivalence relation of P y R N is often areflection of the dynamical properties of P .In addition to P above, consider a Polish permutation group Q on some set M . Noticethat P Wr N Q may be realized as a Polish permutation group on the set N × M via thefaithful action P Wr N Q y N × M with( p, q ) · ( n, m ) ( p · n, ( q ( n )) · m ) . Moreover, its associated generalized Bernoulli shift P Wr N Q y R N × M is naturally iso-morphic to the P -jump P Wr N Q y ( R M ) N of the generalized Bernoulli shift Q y R M of Q . In particular, since E is Borel isomorphic to the orbit equivalence of the (classical)Bernoulli shift of Z , we see that the Z -jump E [ Z ]0 of E can be identified with the orbitequivalence relation of the generalized Bernoulli shift Z Wr Z Z y R Z × Z . In [CC] it was shown that E [ Z ]0 is generically ergodic with respect to E ω . This was latergeneralized in [All] to the fact that E [ Z ]0 is generically ergodic with respect to any orbitequivalence relation of any non-Archimedean TSI Polish group action. The following is aspecial case of Corollary 1.6 (in the form of Lemma 2.4 below). Notice that since E [ Z ]0 hasmeager equivalence classes this implies that it is not classifiable by TSI-group actions. SHAUN ALLISON AND ARISTOTELIS PANAGIOTOPOULOS
Corollary 2.3. E [ Z ]0 is generically ergodic with respect actions of TSI-groups. We proceed to the proof of Corollary 1.6 after restating it in a more precise way:
Lemma 2.4.
Let P ≤ S ( N ) and Q ≤ S ( M ) be Polish permutation groups on countablesets N, M . If
P, Q are non-compact then the closed invariant subspace: { x ∈ R M × N | x ( m, n ) = 0 = ⇒ both orbits [ n ] P ⊆ N and [ m ] Q ⊆ M are infinite } of the Bernoulli shift of ( P Wr N Q ) , has meager orbits and is generically ergodic withrespect to actions of TSI Polish groups.Proof of Corollary 1.6 in the form of Lemma 2.4. It is easy to see that the orbits are mea-ger given that the range( x ) is a countable subset of R , for all x ∈ R N × M .Let M ∞ ⊆ M and N ∞ ⊆ N be the collection of all points whose Q -orbit and P -orbit,respectively, is infinite. Let also Q ∗ ≤ S ( M ∞ ) and P ∗ ≤ S ( N ∞ ) be the image of Q and P respectively in their new representation. It is clear that the closed invariant subspace inthe statement of the corollary is isomorphic to action:( L Wr N K ) y R N ∞ × M ∞ , and as in the previous paragraph this is just the P ∗ -jump of the Benroulli shift of Q ∗ . Butthe Bernoulli shift of Q ∗ is generically ergodic, since Q ∗ is non-compact (see [KMPZ19]),and all orbits of P ∗ are infinite. The rest follows from Theorem 1.5 (cid:3) Corollary 1.6 gives many examples of classification problems which are classifiable bycountable structures but not by actions of TSI groups. We may similarly find orbit equiv-alence relations which are classifiable neither by countable structures nor by TSI groupactions. In fact we may do so while acting with a CLI group.
Proof of Corollary 1.7.
Let l be the additive group of Hilbert space of all sequences ( a n ) n of reals which are square-summable. The action of l on R N with ( a n ) n · ( x n ) n := ( a n + x n ) n is turbulent, see [Gao09]. In particular, it has meager orbits all of which happen to be dense.Let G y X be the Z -jump of l y R N . It is easy to check that this space is turbulent andstill has meager orbits. The rest follows from Theorem 1.5 and the fact that the wreathproduct of CLI groups is CLI. (cid:3) Definable classification induces homomorphism between unbalancedgraphs
The following Theorem is the main result of this section.
Theorem 3.1.
Suppose X is a Polish G -space and Y is a Polish H for Polish groups G and H . For any Baire-measurable homomorphism f : E GX → E HY , there is a comeager set C ⊆ X such that for any x , y ∈ C , if x ! y then f ( x ) ! f ( y ) . For the proof of this Theorem we will rely on two lemmas. The following “orbit-continuity” lemma is essentially [Hjo00, Lemma 3.17] modified as in the beginning of theproof of [Hjo00, Theorem 3.18]. For a direct proof see [LP18].
YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 9
Lemma 3.2.
Suppose X is a Polish G -space and Y is a Polish H for Polish groups G and H . For any Baire-measurable homomorphism f : E GX → E HY , there is an invariantcomeager set C ⊆ X such that:(1) f restricted to C is continuous; and(2) for any x ∈ C , the set of g ∈ G such that gx ∈ C is comeager; and(3) for any x ∈ C and any open neighborhood W of the identity of H , there is an openneighborhood U of x and an open neighborhood V of the identity of G such thatfor any x ∈ U ∩ C and for a comeager set of g ∈ V , we have f ( gx ) ∈ W f ( x ) and gx ∈ C . The next lemma says that the witnesses g x and g y in the definition of x ! y can betaken to be locally generic. Lemma 3.3.
For any G -space X and any x, y ∈ X , if x ! y , then for any open neigh-borhood V of the identity of G and any nonempty open neighborhood U ⊆ X of x or y ,there is a nonmeager set of g x ∈ G and a nonmeager set of g y ∈ G such that g x x, g y y ∈ U , g y y ∈ V ( g x x ) , and g x x ∈ ( V ( g y y ) .Proof. Let U ⊆ X be a nonempty open neighborhood of x or y and V an open neighborhoodof the identity of G . Choose another open neighborhood V of the identity such that V ⊆ V . By the definition we can find some h x , h y ∈ G such that h x , h y ∈ U and h x x ∈ V ( h y y ) and h y y ∈ V ( h x x ).However, observe that for any g x ∈ V h x and g y ∈ V h y , we have g x x ∈ V ( g y y ) and g y y ∈ V ( g x x ). Also, the set of g x ∈ V h x such that g x x ∈ U and the set of g y ∈ V h y such that g y y ∈ U are both open anf nonempty. Thus the sets of g x and g y satisfying theconditions contains nonempty open sets and thus are nonmeager. (cid:3) We turn now to the proof of Theorem 3.1.
Proof of Theorem 3.1.
Fix an arbitrary open neighborhood W of the identity of H , andan open neighborhood U of f ( y ) (the case that U is a neighborhood of f ( x ) is similar).By the orbit-continuity lemma, we can find open neighborhoods U ′ ∋ y and W of theidentity of G such that for every x ∈ U ′ ∩ C , there is a comeager set of v ∈ V such that f ( vx ) ∈ W f ( x ). By shrinking U ′ , we may assume that f [ U ′ ] ⊆ U .Since x ! y , by the previous lemma, we may find some group elements g x , g y ∈ G suchthat g x x , g y y ∈ U ′ ∩ C , g x x ∈ V ( g x x ) and g y y ∈ V ( g y y ). Since f is a homomorphism,we may fix group elements h x , h y ∈ H such that h x f ( x ) = f ( g x x ) and h y f ( y ) = f ( h y y ),which are both elements of U .To see that h x f ( x ) ∈ W ( h y f ( y )), fix an open neighborhood U ∋ h x f ( x ). Notice g x x ∈ f − [ U ] ∩ C and the set v ∈ V such that v ( g y y ) ∈ f − [ U ] is nonempty open,thus we can choose one such that f ( v ( g y y )) ∈ W f ( g y y ) = W ( h y f ( y )). Checking that h y f ( y ) ∈ W ( h x f ( x )) is the same. (cid:3) Strong ergodicity properties and dynamical back and forth
Let X be a Polish G -space. In this section we define a binary relation - αG on X , forevery α < ω . Intuitively, two points x, y ∈ X satisfy x - αG y , if Player II has a non-losingstrategy of rank α in a dynamical analogue of the classical Ehrenfeucht–Fra¨ıss´e game,where Player I is the “spoiler” and starts by partially specifying x . In Proposition 4.1,which is the main result of this section, we derive some strong ergodicity properties underthe assumption x - αG y . Notice that the ideas developed in this section are similar in spiritto the content of [Hjo00, Section 6.4].Let X be a Polish G -space, let V be an open neighborhood of the identity of G andlet x, y ∈ X . For every α < ω we define a relation x - αV y as follows. Let x - V y exactly when x ∈ V y . Now assume for some ordinal α that - βW is defined for every ordinal β < α and every open neighborhood W of the identity of G . Let x - αV y if for every openneighborhood W of the identity of G , there exists some v ∈ V such that for every β < α , vy - βW x . Note that by our choice of notation, these relations are not in general symmetric,but it is also important to note that they are not necessarily transitive either.Before we state the main result of this section, recall the notation associated with theVaught transforms. Let A ⊆ X be a Baire-measurable set and let V ⊆ G be any openset. If x ∈ X , we write x ∈ A ∗ V if the set { v ∈ V | vx ∈ A } is comeager in V . We write x ∈ A ∆ V if the set { v ∈ V | vx ∈ A } is non-meager in V . For basic properties of theVaught transforms one may consult [Gao09]. Proposition 4.1.
Let G an arbitrary Polish group and let X be a Polish G -space. If x - αG y and α ≥ , then for every Π α -set A ⊆ X we have that x ∈ A ∆ G ⇒ y ∈ A ∆ G . We start by recording some useful basic properties of the relations - αV . Lemma 4.2.
Let X be a Polish G -space and let V, W be open neighborhoods of the identityof G . For every α < ω and every x, y, z ∈ X we have that:(1) if x - αV y and g ∈ G , then gx - αgV g − gy ;(2) if x - αV y and y - αW z , then x - αV W z .Proof. For (1), if x - V y , then y = lim n g n x and x = lim n h n y for some g n , h n ∈ V . Butthen, by continuity of the action we have gx = lim n gh n y = lim n gh n g − gy and same for gy ;that is, gx - gV g − gy . Assume now that x - αV y and let W ⊆ G be an open neighborhoodof the identity of G . Since x - αV y and g − W g is an open neighborhood of the identity of G ,there is some v ∈ V such that for every β < α , vy - βg − W g x . By the inductive assumption,for every β < α we have that gvy ∼ βW gx . Hence ( gvg − ) gy ∼ βW gx , as desired.For (2), suppose first that x - V y and y - W z . For an arbitrary open neighborhood U ∋ z , we can find an open neighborhood U ′ ∋ y and some w ∈ W such that mU ′ ⊆ U .Then we can find some v ∈ V such that vx ∈ U ′ , in which case wvx ∈ U , where wv ∈ V W .Thus z ∈ V W x . The fact that x ∈ V W x follows similarly.
YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 11
Now suppose x - αV y and y - αW z for some α ≥
1. Fix an open neighborhood O of theidentity of G , with the goal of showing for some g ∈ V W that gy ∼ βO x , for every β < α .Let O be an open neighborhood of the identity of G so that O ⊆ O . Since x - αV y , thereis v ∈ V with vy - βO x , for all β < α . Now find some w ∈ W such that wz - βv − O v y . ByLemma 4.2.(1), we get vwz - βO vy . By the induction hypothesis, we have vwz - βO x asdesired. (cid:3) We may now proceed to the proof of Proposition 4.1.
Proof of Proposition 4.1.
Notice that if x ∈ A ∆ G , one can find an open neighborhood V ofthe identity of G and a group element g ∈ G such that gx ∈ A ∗ V . By Lemma 4.2.(1), wehave gx - αG gy . Then there is some h ∈ G such that for every β < α , hgy - βW gx , where W is chosen to be some open neighborhood of the identity of G such that W ⊆ V . Thusit suffices to prove the following claim, which tells us that hgy ∈ A ∗ W and thus y ∈ A ∆ G . Claim 4.3.
Let
V, W be open neighborhoods of the identity of G so that W ⊆ V . If forsome β ≥ we have x - βW y and A ∈ Π β +1 ( X ) , then y ∈ A ∗ V implies x ∈ A ∗ W .Proof of Claim. We proceed by induction on β ≥
0. First, suppose x - W y and that y ∈ A ∗ V for some closed set A ⊆ X . Assuming for the sake of contradiction that x A ∗ W ,one can pick some w ∈ W such that w x ∈ U , where U := A c . By Lemma 4.2.(1), we get w x - W w y , in which case one can pick some w ∈ W such that w w y ∈ U . So thereis an open neighborhood of w w of elements w such that wy ∈ U . Since W ⊆ V , thiscontradicts y ∈ A ∗ V .Suppose now that, for some ordinal α ≥
1, the claim is true for all β < α and that x - αW y and y ∈ A ∗ V for some Π α +1 set A . Write A = T n ∈ ω B n for Σ α subsets B n ⊆ X .Assume for the sake of contradiction that x A ∗ W . Then there is some w ∈ W and anopen neighborhood W of the identity of G , as well as some n ∈ ω , such that W w ⊆ W and w x ∈ ( X \ B n ) ∗ W . Choose an open neighborhood W of the identity of G such that W ⊆ W . Then we can find some w ∈ W such that for every β < α , wy - βw − W w x . ByLemma 4.2.(1) we have w wy - βW w x .If α is a successor ordinal, then X \ B n is Π β for some β < α . On the other hand, if α is a limit ordinal, then X \ B n can be written as a countable intersection T m ∈ ω C m whereeach C m is Π β for some β < α , in which case ( X \ B n ) ∗ W = T m ∈ ω C ∗ W m . Thus by theinduction hypothesis applied to w x ∈ ( X \ B n ) ∗ W , we have w wy ∈ ( X \ B n ) ∗ W . But W w w ∩ V = ∅ , contradicting that y ∈ A ∗ V . (cid:3)(cid:3) Dynamics of TSI Polish groups
In this section, we derive some consequences for the relations ! and - αH , when theserelations have been defined on a Polish H -space Y , where H is a TSI Polish group. Wethen conclude with the proof of Theorem 1.3. Throughout this section H is a TSI Polish group and Y is a Polish H -space. We also fixa countable basis B of open, symmetric, and conjugation-invariant neighborhoods of theidentity of H . We assume that G ∈ B and that for any V ∈ B , we have V ∈ B . Lemma 5.1.
Let H be a Polish TSI group and Y a Polish H -space. We have:(1) if x ! y , then x - H y ;(2) - αV is symmetric for all V ∈ B and all ordinals α > ;(3) - αH is an equivalence relation for all ordinals α > .Proof. For (1), notice that if g y y = lim n v n g x x for some g x , g y ∈ H and some sequence ( v n )from some V in B , then v n g x = g x w n for some sequence ( w n ) in V , and therefore gy - V x ,for g := ( g x ) − g y . For (2), if x - αV y and W ∈ B , then we have that vy - βW x for all β < α . By inductive hypothesis we have x - βW vy , and by Lemma 4.2 (1) it follows that v − x - βW y for all β < α . Hence, y - αV x . For (3), transitivity follows from Lemma 4.2(2),and symmetry by (2) above. (cid:3) In the rest of this section we will want to refer to the relations - αV computed accordingto multiple Polish topologies on the same space. For any topology σ making ( Y, σ ) a Polish H -space, we will use the notation - α,σV to refer to the relation - αV as computed in thatspace. We will use τ to refer to the original topology on Y , but keep denoting - α,τV simplyby - αV . For every V ∈ B , c ∈ Y , and ordinal β , let A βV ( c ) := { d ∈ Y | d - βV c } be the downward cone of c with respect to - βV . We have the following lemma. Lemma 5.2.
Let σ be an additional topology on Y so that both Y and ( Y, σ ) are Polish H -spaces. Let a, b, c ∈ Y and let α ≥ be an ordinal so that: for every β < α we have a - βH c and b - βH c ; and for all V, W ∈ B , the set ( A βV ( C )) ∆ W is in σ . Then a - ,σH b = ⇒ a - αH b. Proof.
Let V be an arbitrary open neighborhood of the identity of H . Our goal is to showthat there is some h ∈ H such that for every β < α , hb - βV a . To that end, let V ∈ B with V − ⊆ V , and fix h ∈ H such that hb - ,σV a . Fix any ordinal β such that β < α . Weclaim that hb - βV a . To see this, let W ∈ B . We will find some v ∈ V such that va - γW hb ,for all γ < β .Fix any γ < β . Let W ∈ B so that W ⊆ W . By Lemma 5.1(2) and assumptionwe may choose some g ∈ H such that ga - γW c , in which case ga ∈ A γW and thus, byLemma 4.2(2), a ∈ ( A γW ) ∆ W g . Since ( A γW ) ∆ W g is an open neighborhood of a in σ ,there is v ∈ V such that vhb ∈ ( A γW ) ∆ W g . By definition, for some w ∈ W , we have wgvhb - γW c . Since ga ∈ A γW and wgvhb - γW c , by Lemma 4.2(2) and Lemma 5.1(2), wehave that ga - γW wgvhb , and thus ga - γW gvhb . By Lemma 4.2(1), we get v − a - γW hb as desired. (cid:3) YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 13
We may now proceed to the proof of Theorem 1.3.
Proof Theorem 1.3.
Let f : X → Y be a Baire-measurable homomorphism from E GX to E HY , for some Polish H -space Y , where H is a TSI Polish group. Claim 5.3.
For all α < ω there is comeager C α ⊆ X so that for all x, y ∈ C α f ( x ) - α +1 H f ( y ) . Proof.
For α = 1, by Lemma 3.1 we have a comeager set D such that for any x, y ∈ D , if x ! y , then f ( x ) ! f ( y ). Since ( X/G, ! ) is generically semi-connected, we can finda comeager set C ⊆ D such that for any x, y ∈ C , there is a ! -path between x and y through D . In particular, by Lemma 5.1(3), for any x, y ∈ C , we have f ( x ) - H f ( y ). Sowe may set C := C .Assume now that for some countable α ≥ C β for all β < α asin the claim. Notice that for any fixed z ∈ T β<α C β and β < α , the set A βV ( f ( z )) = { y ∈ Y | y - αV f ( z ) } is Borel. Thus we may find a new topology σ on Y such that: ( A βW ) ∆ V isopen for every β ≤ α and V, W ∈ B ; (
Y, σ ) is a Polish H -space; and σ generates the sameBorel sets as τ (see Lemma 4.4.3 of [Gao09]). Arguing as in the last paragraph with thespace ( Y, σ ) in place of Y , we can find a comeager set C ⊆ T α<β C α and some z ∈ C sothat for every x ∈ C , f ( x ) - σ, H f ( z ). By Lemma 5.2 we have that f ( x ) - αH f ( y ), for every x, y ∈ C . Set C α := C . (cid:3) Fix now a countable ordinal λ and a comeager set C ⊆ X such that for any x ∈ D , theorbit of f ( x ) is Π λ . By the previous claim, the set D := C ∩ C λ is comeager and for any x, y ∈ D , f ( x ) - λH f ( y ). By Lemma 4.1, this means that for any Π λ set A , f ( x ) ∈ A ∆ H iff f ( y ) ∈ A ∆ H . Since [ f ( x )] is Π λ for all x ∈ D , we have that every x ∈ D maps to thesame H -orbit in Y . (cid:3) Applications
In this section we illustrate how the “in vitro” results we have developed so far ap-ply to natural classification problems from topology and operator algebras. We start byreviewing some definitions regarding fibre bundles. We then show that coordinate free iso-morphism between Hermitian line bundles and
Morita equivalence between continuous-trace C ∗ -algebras are not classifiable by TSI-group actions.6.1. The Polish space of locally trivial fibre bundles.
Let B be a locally compactmetrizable topological space, and let F be a Polish G -space, for some Polish group G . Alocally trivial fibre bundle over B with fibre F and structure group G , or simply a fibrebundle over B consists of a Polish space E ; a continuous map p : E → B ; a locally finiteopen cover U of B ; and a homeomorphism h U : p − ( U ) → U × F , for each U ∈ U ; so that:(1) if b ∈ U ∈ U , then h U restricts to a homeomorphism from p − ( b ) to { b } × F ;(2) if U, V ∈ U , there is a contiunous t ( U,V ) : U ∩ V → G , so that for all b ∈ U ∩ V ,( h V ◦ h − U )( b, f ) = ( b, t ( U,V ) ( b ) f ) The maps h U above are called charts and t ( U,V ) are called the transition maps . Noticethat we can always choose U to be a subset of some fixed countable basis B of the topologyof B , and we can recover E as the colimit of the above separable data (together with a1-cocycle condition). Hence, we may form the Polish space Bun( B, G, F ), of all locallytrivial fibre bundle over B , with fibre F , and structure group G , as a G δ subset of thePolish space 2 B × Y ( U,V ) ∈B C ( U ∩ V, G )There are two natural classification problems on Bun(
B, G, F ): the isomorphism rela-tion ≃ iso ; and the isomorphism over B relation ≃ B iso . First, notice that If p, q : E → B are elements of Bun( B, G, F ), then we may always choose a common open cover U of B sothat p and q are locally trivialized by some ( h U ) , ( t ( U,V ) ) and ( k U ) , ( s ( U,V ) ), respectively,with U, V ∈ U . We write p ≃ iso q , if there are homeomorphisms π : E → E , ρ : B → B ,and a continuous e ( U,V : U ∩ ρ − ( V ) → G , so that q ◦ π = ρ ◦ p , and for all U, V ∈ U , forall b ∈ U ∩ ρ − ( V ), and for all f ∈ F we have that( h V ◦ π ◦ h − U )( b, f ) = (cid:0) ρ ( b ) , e ( U,V ) ( b ) f (cid:1) . We write p ≃ B iso q if ρ above can be taken to be id B .6.2. Isomorphism of Hermitian line bundles.
Let B be a locally compact metrizablespace. By a Hermitian line bundle over B we mean any locally trivial fibre bundleover B with fibre F := C and structure group G := U( C ) = T being the unitary group of C acting on C with rotations. Let Bun C ( B ) be the standard Borel space of all Hermitianline bundles over B . By a result of [BLA19], ≃ B iso is classifiable by TSI group actions: Proposition 6.1 ([BLA19](Corollary 5.12.)) . The problem (Bun C ( B ) , ≃ B iso ) is classifiableby non-Archimedean, abelian group actions. In contrast, for the relation ≃ iso we have the following result. Corollary 6.2.
There exists a locally compact metrizable topological space B , so that (Bun C ( B ) , ≃ iso ) is not classifiable by TSI group actions. In fact, B can be taken to be thegeometric realization of a countable, locally-finite, CW-complex.Proof. The dyadic solenoid Σ is the inverse limit of the inverse system ( T i , f ji ) where T i := T is the unit circle, viewed as a multiplicative subgroup of C , and f ji : T j → T i is thetwo-fold cover z z . Let C be the homotopy limit of the same inverse system. This isformed by taking the disjoint union of the spaces: T × [0 , , T × [0 , , T × [0 , , . . . and identifying the point ( z, ∈ T i +1 × [0 ,
1] with the point ( z , ∈ T i × [0 , i ≥
0. Clearly C is a locally finite CW-complex.Recall now that the quotient Bun C ( C ) / ≃ C iso is in bijective correspondence with thefirst ˇCech cohomology group H ( C, T ) of C with coefficients from T ; [RW98, Proposition4.53]. Utilizing the short exact sequence 0 → Z → R → T → YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 15 covering of T the later is isomorphic to the second ˇCech cohomology group H ( C, Z ) of C with coefficients from Z [RW98, Theorem 4.42]. By Steenrod duality [Ste40], and since C is homotopy equivalent to a solenoid complement S \ Σ, H ( C, Z ) isomorphic to 0-thSteenrod homology group H ( C, Z ).In [BLA19] it was shown that the ˇCech cohomology groups for locally compact metriz-able spaces, as well as the Steenrod homology groups for compact metrizable spaces, arequotients of Polish G -spaces. Moreover all the computations described in the previousparagraph lift to Borel reductions on the level of Polish spaces; see [BLA19, Lemma 2.14,Theorem 3.12, and Section 5.5]. By [BLA19, Proposition 4.2] we have (Bun C ( C ) , ≃ C iso ) isBorel bireducible with the orbit equivalene relation of the action of Z on its dyadic profi-nite completion Z by left-translation. Moreover, by Borel functoriality of the definableˇCech cohomology—this is proved in [BLP], but for CW-complexes it can be checked byhand—the action of Homeo( C ) on C induces definable endomorphisms of0 → Z → Z → Z / Z → , as in [BLA19, Section 5.3]. If follows by [BLA19, Proposition 5.6] that (Bun C ( C ) , ≃ iso ) isBorel bireducible to eventual equality of binary sequences (2 N , E ).Fix some point p in C , say the one corresponding to (1 , ∈ T × [0 , B be theCW-complex which is attained by taking the disjoint union of Z -many copies ( C k ) of C and connecting the point p of C k to the point p of C k +1 by gluing on them the endpointsof a homeomorphic copy of the interval [0 , B is a Z -line of intervals. Everyhomeomorphism of C acts on the indexing copy of Z by the group Γ := ( Z / Z ) ⋊ Z in theobvious way. It is easy to see that the Γ-jump E [Γ]0 of E reduces to (Bun C ( B ) , ≃ iso ). ByTheorem 1.3 and Theorem 1.5 we have that E Γ0 is not classifiable by TSI-group actions. (cid:3) Morita equivalence of continuous-trace C ∗ -algebras. In what follows we willonly consider separable C ∗ -algebras A whose spectrum b A is Hausdorff. This implies that b A is a locally compact metrizable space. By the Gelfand-Naimark theorem, the subclassof all such commutative C ∗ -algebras is “locally-concretely” classified via the assignment A b A : every two commutative C ∗ -algebras with homeomorphic spectrum are isomorphic.The unique up to isomorphism such commutative C ∗ -algebra of spectrum S is simply thealgebra C ( S, C ), of all continuous maps from S to C which vanish at infinity. It turns outthe Borel complexity of similar “local” classification problems increases drastically even inthe case of continuous-trace C ∗ -algebras, which is the closest it gets to being commutative.For more on the general theory of C ∗ -algebras than we provide here, see [RW98, Bla06].Let S be a locally compact metrizable space and let K ( H ) be the C ∗ -algebra of allcompact operators on the separable Hilbert space. Two C ∗ -algebra A, B with spectrum S are Morita equivalent if A ⊗ K ( H ) and B ⊗ K ( H ) are isomorphic as C ∗ -algebras. Ingeneral, this isomorphism may only preserve the spectrum up to homeomorphism. Whenthis induced homeomorphism can be taken to be id S then we say that A and B are Moritaequivalent over S . Any C ∗ -algebra A with Hausdorff spectrum can be endowed with C ( S )-module structure, where C ( S ) is the collection of all continuous f : S → C which vanish at infinity. As a consequence, A and B are Morita equivalent over S if and only ifthey are isomorphic via a C ( S )-linear map.Let CTr ∗ ( S ) be the space of all continuous-trace C ∗ -algebras with spectrum S .These are all C ∗ -algebras A , for which there is an open cover U of S consisting of relativelycompact sets so that, for all U ∈ U , if A U is the quotient algebra induced by U ⊆ S ,then A U is Morita equivalent to C ( U , C ) over U ; [RW98, Proposition 5.15]. In other wordscontinuous-trace C ∗ -algebras are precisely the algebras which are locally Morita equivalentto commutative. It turns out that any algebra A ∈ CTr ∗ ( S ) can be identified with a locallytrivial fibre bundle over S , whose fibre F is the C ∗ -algebra K ( H ) of all compact operatorsof a separable (potentially finite dimensional) Hilbert space, and the strcture group G is Aut( K ( H )); [Bla06, IV.1.7.7, IV.1.7.8]. Hence, similarly to subsection 6.1, we mayview CTr ∗ ( S ) as a Polish space. The space CTr ∗ Stable ( S ) of all stable continuous-trace C ∗ -algebras with spectrum S is the subspace Bun( B, Aut( K ( H )) , K ( H )) of CTr ∗ ( S )—where H is the separable infinite dimensional Hilbert space.We consider the following two classification problems: let (CTr ∗ ( S ) , ≡ M ) be the problemof classifying all elements of CTr ∗ ( S ) up to Morita equivalence; and let (CTr ∗ ( S ) , ≡ S M ) bethe problem of classifying all elements of CTr ∗ ( S ) up to Morita equivalence over S . By aresult of [BLA19], ≡ S M is classifiable by TSI group actions: Proposition 6.3 ([BLA19](Corollary 5.14.)) . The problem (CTr ∗ ( S ) , ≡ S M ) is classifiableby non-Archimedean, abelian group actions. In contrast, for the relation ≡ M we have the following result. Corollary 6.4.
There exists a locally compact metrizable topological space S , so that (CTr ∗ ( S ) , ≡ M ) is not classifiable by TSI group actions. In fact, S can be taken to bethe geometric realization of a countable, locally-finite, CW-complex.Proof. First notice that the Borel map implementing A A ⊗ K ( H ), is a Borel reduc-tion, inducing a bijection from CTr ∗ ( S ) / ≡ M to CTr ∗ Stable ( S ) / ≡ M . Hence, it sufffices toconsider the problem (CTr ∗ Stable ( S ) , ≡ M ) instead.By the Dixmier-Douady classification theorem we have that CTr ∗ Stable ( S ) / ≡ S M is inbijective correspondence with the third ˇCech cohomology group H ( S, Z ) of S with coeffi-cients from Z ; see [RW98, Theorem 5.29]. It follows that CTr ∗ Stable ( S ) / ≡ M is in bijectivecorrespondence with H ( S, Z ) / Γ, where Γ is the group of all automorphism of H ( S, Z )induced by the action of Homeo( S ) on H ( S, Z ); see [Bla06, IV.1.7.15]. Similarly to theproof of Corollary 6.2, the Dixmier-Douady correspondence and all the cohomological ma-nipulations lift to Borel reductions on the level of a appropriate Polish spaces; see [BLA19].The rest of the proof follows as in Corollary 6.2: let D be the suspension C × [0 , / ∼ ofthe space C which we defined in the proof of Corollary 6.2, and let S be attained from D in the same way that B was attained from C in the same proof. Notice that by propertiesof the suspension we have that H ( D, Z ) = H ( C, Z ). (cid:3) Remark 6.5.
The complexity of the (CTr ∗ ( S ) , ≡ M ) has been studied in [BLP] for severalspaces S . For example, it is shown that, when S is the homotopy limit coming from YNAMICAL OBSTRUCTIONS FOR CLASSIFICATION BY ACTIONS OF TSI GROUPS 17 the defining inverse system of a d -dimensional solenoid, then (CTr ∗ ( S ) , ≃ Mor ) is alwaysessentially countable but, when d ≥
2, it is not essentially treeable.7.
The space between TSI and CLI
With Corollary 1.7 we established that the class of CLI Polish groups can produce strictlymore complicated orbit equivalence relations than the class of TSI groups from the pointof view of Borel (or even Baire-measurable) reductions. The obvious question is how manydifferent complexity classes lie between the class of all classification problems which areclassifiable by TSI-group actions and the ones which are classifiable by CLI-group actions.In this final section we illustrate how the methods we developed here can be adapted toshow that there is an ω -sequence of strictly increasing complexity classes. The discussionhere will be informal since the details will be provided in an upcoming paper.Let X be a Polish G and recall the unbalanced graph relation ! that we definedbetween pairs of points of X . In the context of the next definition we may refer to it asthe 1 -unbalanced relation and we denote it by ! G . Definition 7.1.
Let X be a Polish G -space, let V be an open neighborhood of the identityof G , and let α < ω . We define the relation ! αV on X by induction. Let(1) x ! V y , if y ∈ V x and x ∈ V y ;(2) x ! αV y , if for every open neighborhood W of the identity of G , and every openneighborhood U ⊆ X of x or y ,there exist v x , v y ∈ V with v x x ∈ U and v y y ∈ U ,so that v y y ! βV v x x , for all β < α .The α -unbalanced graph associated to G y X is the graph ( X/G, ! αG ). CLITSI α -balancedIn [Mal11], Malicki used iterated Wreath prod-ucts to define an ω -sequence ( P α | α < ω ) ofPolish permutation groups. Using this sequencehe established that the colleciton of all CLI groupforms a coanalytic non-Borel subset of the stan-dard Borel space of all Polish groups. Using thetechniques we developed here one may show thatthe α -unbalanced graph of the Bernoulli shift of P α is generically semi-connected (see Section 1.2) andthat any orbit equivalence relation with genericallysemi-connected β -unbalanced graph is genericallyergodic for actions of P α , when α < β . Moreover,in a certain weak sense, these complexity classesare cofinal in the class of all orbit equivalence rela-tions of CLI groups: if for any pair x, y of elementsof a Polish G -space X we have x ! αG y for allcountable ordinals α , then G cannot be CLI. References [All] S. Allison. Actions of tsi closed subgroups of the symmetric group. In preparation.[Bla06] Bruce Blackadar.
Operator algebras. Theory of C*-algebras and von Neumann algebras , volume122 of
Encyclopaedia of Mathematical Sciences . Springer, 2006.[BLA19] J. Bergfalk, M. Lupini, and Panagiotopoulos A. Definable (co)homology, pro-torus rigidity, and(co)homological classification. https://arxiv.org/abs/1909.03641 , 2019.[BLP] J. Bergfalk, M. Lupini, and A. Panagiotopoulos. Definable homology and cohomology and clas-sification problems. In preparation.[CC] J. Clemens and S. Coskey. New jump operators on equivalence relations, and scattered linearorders. In preparation.[Gao09] S. Gao.
Invariant descriptive set theory , volume 293 of
Pure and Applied Mathematics . CRCPress, 2009.[Hjo00] G. Hjorth.
Classification and orbit equivalence relations , volume 75 of
Mathematical Surveysand Monographs . American Mathematical Society, 2000.[Hjo05] G. Hjorth. A dichotomy theorem for being essentially countable. In
Logic and its applications ,volume 380 of
Contemp. Math. , pages 109–127. Amer. Math. Soc., 2005.[KMPZ19] A. S. Kechris, M. Malicki, A. Panagiotopoulos, and J. Zielinski. On Polish groups admittingnon-essentially countable actions. 2019.[LP18] M. Lupini and A. Panagiotopoulos. Games orbits play and obstructions to Borel reducibility.
Groups Geom. Dyn. , 12:1461–1483, 2018.[Mal11] M. Malicki. On Polish groups admitting a compatible complete left-invariant metric.
J. SymbolicLogic , 76:437–447, 2011.[Neu76] P. M. Neumann. The structure of finitary permutation groups.
Arch. Math. (Basel) , 27:3–17,1976.[RW98] I. Raeburn and D.P. Williams.
Morita equivalence and continuous-trace C*-algebras , volume 60of
Mathematical Surveys and Monographs . American Mathematical Society, 1998.[Ste40] N.E. Steenrod. Regular cycles of compact metric spaces.
Annals of Mathematics , 41(4):833–851,1940.
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213
E-mail address : [email protected] URL : Mathematics Department, Caltech, 1200 E. California Blvd, Pasadena, CA 91125
E-mail address : [email protected] URL ::