Continuity properties of measurable group cohomology
A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood.
This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to ‘regularize’ measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument.
First, we prove that for target modules that are Fréchet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fréchet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules.
Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.
Msc:20J06 22C05 18G99 37A99
- 1 Introduction
- 2 Preliminaries
- 3 Improving the regularity of cocycles
- 4 Fréchet modules and continuous cocycles
- 5 Behaviour under inverse and direct limits
1.1 Cohomology for locally compact groups
The cohomology of discrete groups came into being in the works EilMac47-1; EilMac47-2 of Eilenberg and MacLane. It emerged from Hurewicz’ classical discovery that the cohomology groups of two aspherical simplicial complexes are equal if those complexes have isomorphic fundamental groups Hur36, and it then quickly developed in papers of Eilenberg, MacLane, Hopf, Eckmann and others.
In addition to clarifying the structure of these invariants from algebraic topology, this theory has proved useful within group theory in various ways. On the one hand, the low-degree (degree 1 and 2) cohomology classes were found to correspond with naturally-defined data describing crossed homomorphisms and Abelian extensions of groups, and so enabled a streamlined understanding of those data. On the other, cohomology classes in various degrees can be associated to a wide range of different kinds of action of a group, and so can help in understanding those actions. In particular, the theory often allows the issue of whether a given action admits some additional structure to be boiled down to its simplest possible residue in the form of a functional equation which may or may not have a solution. This ability to treat obstructions systematically has led to further points of interaction with both algebraic topology and number theory.
These aspects of the cohomology of abstract groups are by now described in a range of standard texts, such as Brown’s thorough and accessible book Bro82. Since the 1950s group cohomology has also taken its place as a central motivating example within the more abstract study of homological algebra, for which we recommend Weibel’s treatment Wei94. In addition, introductions to various instances of interplay between group theory and topology that result from cohomology theory can be found in Bredon Bre67 and Thomas Tho86.
Almost immediately, the question poses itself of how to define a similar theory for topological groups and topological -modules , and to reap the same benefits for topological groups as in the case of discrete groups.
We consider an Abelian topological group upon which acts as a topological transformation group of automorphisms. One issue is what kinds of topological group to consider. The most natural choice for is a locally compact group, and we will restrict ourselves here to that class, and in addition will always assume that any locally compact group discussed satisfies the second axiom of countability. At first sight it is natural to assume that -modules should also be locally compact, but it is essential both for applications and for the coherence of the theory that this class be expanded to the class of Polish -modules (a Polish topological group is one that admits a complete and separable metric). In addition to all locally compact -modules, this class includes all separable Banach and Hilbert spaces, plus a significant class of other separable topological vector spaces (the F-spaces). It also includes important function spaces such as the space of all measurable functions from a standard Borel -finite measure space into a Polish group , where this is given the topology of convergence in measure. Also, a countable product of Polish groups is Polish.
We denote the category of Polish -modules by . Exact sequences in are sequences
which are exact algebraically with and continuous. It follows from standard results on Polish groups Ban55 that is automatically a homeomorphism onto its range and that automatically induces a homeomorphism of the quotient group with , so that the sequence is exact in a very strong sense.
What is sought in this context is a family of covariant functors , , from to Abelian groups with , the subgroup of -fixed points in , and so that to every exact sequence (1) in there are dimension-shifting connecting homomorphisms (sometimes called ‘switchbacks’) so that everything fits together to form an infinite exact sequence of cohomology
Call such a family of functors, a cohomological functor . Finally, we want these functors to be effaceable in , which means that for any and any there is an exact sequence (1) so that the image of in vanishes. By a well-known argument of Buchsbaum Buc60, an effaceable cohomological functor on , if one can be found, must be unique. Importantly, unlike discrete group cohomology, such an effaceable cohomological functor cannot be obtained by computing derived functors from injective resolutions, because the category generally does not have enough injectives.
One of us introduced in a series of papers beginning in 1964 Moo64(gr-cohomI-II); Moo76(gr-cohomIII); Moo76(gr-cohomIV) cohomology groups satisfying these requirements, based on a complex of measurable cocycles from into (or more properly equivalence classes of measurable functions which agree almost everywhere) with the usual coboundary operator. (Precise definitions will be recalled in Section 2 below.) This leads to cohomology groups we denote by (‘m’ standing for ‘measurable’). One could also use Borel functions, but this leads to the same end result. These groups have the right values in degree , and the dimension-shifting connecting maps always exist and fit into a long exact sequence of cohomology corresponding to any short exact sequence (1). Finally they are effaceable, and hence form the unique effaceable cohomological functor on .
In addition, the groups have a natural topology which comes from the quotient structure where , the group of measurable cocycles, is itself Polish and where is the continuous image of another Polish group. If the latter group happens to be closed in the former group, or equivalently if the quotient is Hausdorff, then is also a Polish group. However, the coboundaries are not always closed and in this case still has a topology, but it is not particularly useful.
Although is the unique effaceable cohomological functor on , many different candidates for a cohomology theory of topological groups have emerged over the years based on other sets of requirements. We will now briefly discuss five of these candidates.
The first appearance of cohomology groups of topological groups was in class field theory, where Galois groups of infinite field extensions and their cohomology groups appeared naturally. These Galois groups are profinite and so compact and totally disconnected, and the cochains that appeared naturally were continuous ones (see ArtTat51 and also ArtTat09). This cohomology theory for infinite Galois groups and its applications to class field theory were developed by Tate and his students. At almost the same time van Est began a study of the cohomology of Lie groups acting on finite dimensional vector spaces, again using a complex of continuous cochains vEs53. This work was continued and extended by Hochschild and Mostow Mos61; Hoc61; HocMos62 and then developed further by Borel and Wallach BorWal00, who specifically included general infinite-dimensional Fréchet spaces as -modules.
In general, for any and any Polish -module , one can introduce the cochain complex of continuous functions from to with the usual coboundary operator on cochains. This is the most natural and straightforward generalization from the case of discrete groups, and produces a family of functors that we denote . They satisfy and are effaceable. In addition, if is totally disconnected and is arbitrary, or if is arbitrary and is restricted to the subcategory of modules that are Fréchet spaces, then the dimension-shifting connecting maps do exist and there is a long exact sequence of cohomology. In both cases this is a consequence of results giving continuous lifts for short exact sequences of modules under these assumptions111In fact, earlier works such as Hochschild and Mostow’s HocMos62 simply narrowed the requirement for long exact sequences to only those short exact sequences of modules that admit continuous left-inverses as sequences of topological spaces. With this convention, the theory always has long exact sequences, but on the other hand the theory is not effaceable in if one allows only inclusions of modules that give rise to such distinguished short exact sequences.. These are the two cases where these groups were introduced and developed very successfully. However, for general and and short exact sequences as in (1) the dimension-shifting connecting maps for are missing, so there is not always a long exact sequence of cohomology. We note that, as for , the groups inherit a topology as a quotient of the -cocycles (with the compact-open topology) modulo the -coboundaries, which also may or may not be Hausdorff.
In 1973 Wigner Wig73 introduced two further cohomology theories. The first is based on equivalence classes of multiple extensions of elements of , as studied by Yoneda Yon60 in the non-topological case. The multiple extensions in dimension are exact sequences of the form
where the are in and where is given the trivial -action. This construction leads to cohomology groups (‘’ for ‘Yoneda-Wigner’) which are easily seen to be effaceable cohomological functors with the correct value for . Hence by the uniqueness theorem, Wigner concludes that for all and ; we will not discuss further in this paper.
Secondly, Wigner adapts a construction of Grothendieck, Artin, Verdier, and Deligne to this topological context. Wigner builds a semi-simplicial space over , which at level is , together with a semi-simplicial sheaf corresponding to any -module in . The sheaf at level consists of germs of continuous functions from into . One then resolves this semi-simplicial sheaf to get cohomology groups (‘’ for ‘semi-simplicial’). Wigner shows these are effaceable functors with the right value for . However, the dimension-shifting connecting maps do not always exist. It is easy to see that they do exist if the exact sequence of -modules (1) has continuous local cross-sections from back up to , but without this assumption a short exact sequence of modules may not correspond to a short exact sequence of sheaves on . One can construct long exact sequences in for short exact sequences of sheaves, but one has an adequate supply of short exact sequences only after enlarging the class of permitted sheaves on to examples not arising from members of .
Wigner’s main result here is that if is finite-dimensional and if is a member of a certain subcategory of , consisting of Polish modules that have something he calls ‘Property F’, then connecting maps always exist using only the sheaves constructed from objects of ; and by the Buchsbaum uniqueness theorem applied to the category , it follows that for finite-dimensional and in . (One also has to check that both of these theories are effaceable within the subcategory, which is the case.) Later we will give an example of and where is not the same as , so Wigner’s restrictions are not superfluous. We note that the cohomology theory has been further developed and refined in recent papers of Lichtenbaum Lic09 and Flach Fla08, working via a more general topos-theoretic redefinition based on ideas of Grothendieck, Artin and Verdier SGA4t1; SGA4t2; SGA4t3 which always gives the same theory as Wigner’s in his setting. Those more recent works are motivated by number-theoretic applications to the interpretation of values of the Dedekind Zeta-functions of number fields.
Graeme Segal in 1970 Seg70 developed another cohomology theory of topological groups using a type of contractible resolution of a -module . To be comparable, and have to be k-spaces, which they are in our case; all -modules have to be locally contractible; and only those exact sequences of -modules which have topological local cross-sections are allowed. Unlike the category of Polish -modules, this category admits a natural family of canonical resolutions of such modules, and using these Segal defines his theory much as the cohomology of classical groups can be obtained as a sequence of derived functors. We call this theory , and it is clear that in the context where they are both defined one has (see Seg70 Section 3).
The last cohomology theory for topological groups we mention is a direct generalization of the topological interpretation of cohomology for discrete groups in terms of classifying spaces. For a topological group one may construct a locally trivial principal -bundle with contractible, universally up to homotopy, and having done so the base is called a classifying space for . See, for instance, Husemöller Hus95, and also the monograph HofMos73 of Hofmann and Mostert. In our case these spaces may be taken to be paracompact. Then for any in , one can form a locally trivial associated fibre bundle over with fibre , and then form the sheaf of germs of continuous sections of this bundle; see Seg70 Proposition 3.3. The sheaf cohomology groups are possible candidates for cohomology groups of . However there is a problem in that is defined only up to homotopy type and sheaf cohomology may not be homotopy-invariant unless the sheaf is locally constant. This means that for a well-defined theory we need to restrict to discrete , so that is locally constant. (Although having fixed a choice of the groups for more general do have an auxiliary rôle in Segal’s work: he uses dimension-shifting to construct a comparison , and so he cannot stay within the class of discrete modules.) For a discrete -module , we denote (‘’ for ‘classifying space’). For discrete these coincide with the usual cohomology for discrete groups, since in this case is a . Segal Seg70 shows that and agree when both are defined, that is for discrete . Wigner also proves that and agree in this case.
While all of these theories play important rôles, clearly and have the widest scope in terms of permissible coefficients and groups , and our primary focus here will be on and on expanding the area of its agreement with . Various analytic and computational questions about these cohomology groups remained open following Moo64(gr-cohomI-II); Moo76(gr-cohomIII); Moo76(gr-cohomIV), and the present paper resolves some of these. Our principal results are of three kinds:
sufficient conditions for the cohomology bifunctors to be continuous under inverse or direct limits of the arguments;
sufficient conditions for the cohomology groups to be Hausdorff in their quotient topologies;
and conditions under which can be shown to agree with another of the theories discussed above.
As we shall see, the proofs of these results have certain key analytic arguments in common.
Before proceeding further, let us offer another general note concerning measurable cohomology . It is reasonable to ask why this cohomology theory for topological groups and based on measurable cochains, which themselves appear to be so very weakly linked to the topological structure of and , should work at all. Two points should be made. First, the topology on a locally compact group can be constructed or reconstructed from the measure theory — that is, from the Borel structure and an invariant or quasi-invariant measure on . Mackey established this in Mac57 by refining an earlier result of Weil (which first appeared in Wei64). Hence the measure theory on determines the topology on in a rather strong sense. Second, one has a variety of automatic continuity theorems for Polish groups such as the following. If is Polish group and is a homomorphism into a separable metric group which is a Borel function (in that the inverse image of every Borel set in is a Borel set in ), then is continuous (p.23 in Ban55). So again for Polish groups , the Borel structure has a lot to say about the topology of .
1.2 Comparison with continuous cochains
A major concern of this paper will be to expand the areas of agreement between , , , and . The first two of these are defined simply in terms of cochains (measurable or continuous, respectively) in a bar resolution, and so their comparison will use only more elementary arguments.
Simple examples show that these theories can differ, and the continuous-cochains theory lacks both the universality properties of when considered on the whole of and also the correct interpretation in terms of group extensions in degree (rather, it captures those group extensions that admit global continuous sections). However, it has long been suspected that these theories do coincide in the following situation.
Theorem A If is a locally compact second countable group and is a Fréchet -module then the natural comparison homomorphism
from continuous cohomology is an isomorphism: that is, every class in has a continuous representative in the bar resolution, and if the class is trivial then that continuous representative is the coboundary of another continuous cochain.
If in addition is compact then for all .
Note that the analogous agreement is given by Theorem 3 of Wig73.
If is a Fréchet module and is compact, then given a bounded measurable cocycle a standard averaging trick shows that it is a coboundary. We prove Theorem A below using the same idea, but before it can be applied one must show that all cohomology classes have measurable-cocycle representatives that are ‘locally bounded’ in a suitable sense. This follows from an elementary procedure for regularizing measurable cocycles based on dimension-shifting, which is the first innovation of the present paper. A similar reduction to bounded cocycles already appeared in the proof of Theorem 2.3(1) of Moo64(gr-cohomI-II) concerning , but that relied instead on the interpretation of this group in terms of group extensions together with deep results of Mackey and Weil on the structure of standard Borel groups. By contrast, the proof of Theorem A below uses only elementary analysis of cocycles themselves.
1.3 Continuity properties and inverse and direct limits
Another aspect of the groups that has remained unclear is their behaviour under forming inverse limits in the first argument or direct limits in the second. If is a continuous epimorphism of locally compact second countable groups and is a Polish -module (which we identify also as a -module by composing with the epimorphism), then the natural inflation maps are a sequence of homomorphisms . As a result, if , is an inverse system of locally compact second countable groups with a locally compact second countable inverse limit , , and is a module for all these groups (say, lifted from a module for some minimal group in the system), we may form the direct limit of the associated inflation maps to obtain a chain of homomorphisms
A natural question is whether these are isomorphisms. It is clear that this is not always so: for example, if is a sequence of finite-dimensional compact Abelian groups converging to an infinite dimensional compact Abelian group and we set with the trivial action of each , then the identity map is a Borel crossed homomorphism, hence a -cocycle, but it clearly is not lifted from any of the groups ; and in this case there are no -coboundaries, so the cohomology class of this identity element contains no other cocycles, and hence that class too is not lifted from any .
However, in this paper we show that for compact base groups and for certain large classes of target module the above continuity under inverse limits does in fact obtain.
Theorem B If , is an inverse system of compact groups with second countable inverse limit , then
under the direct limit of the inflation maps whenever is a discrete Abelian group or a finite-dimensional torus with an action of that factorizes through every .
Remark Karl Hofmann has pointed out to us that there are many naturally-occurring functors of topological algebra that respect inverse limits of compact groups, as above, but do not respect more general categorial limits. It is quite possible this situation holds for , but we have not examined this question. These issues are discussed very generally in Hof76.
Since a torus is a quotient of a Euclidean space by a lattice, cohomology for discrete and toral targets can be connected using the long exact sequence of cohomology and an appeal to Theorem A. The heart of Theorem B is therefore the case of discrete , and this relies on a more quantitative version of our basic cocycle-smoothing procedure.
Similarly to the above, if now is fixed, is a discrete -module and is some direct system of discrete submodules with union the whole of , then these inclusions define a system of homomorphisms
and once again we may ask whether these are isomorphisms. In fact the same basic estimates as needed for Theorem A show that this is often also the case.
Theorem C If is any compact second countable group, a discrete -module and a direct system of submodules such that then
under the direct limit of the maps on cohomology induced by the inclusions .
Remark Simply by applying Theorem C and then Theorem B in turn, one can deduce at once a strengthening of Theorem B to the case when is an arbitrary discrete Abelian -module, according to which
where denotes the submodule of elements of that are individually fixed by the kernel of , which may be re-interpreted as a -module. The applicability of Theorem C to this situation follows because the orbits of the -action on must be compact, hence finite, and so every element of is fixed by some : that is, .
A few low-degree cases of the above results already appear as Theorems 2.1, 2.2 and 2.3 in Part I of Moo64(gr-cohomI-II), also proved using the group-theoretic identification of the elements of with locally compact extensions of by .
For non-compact base groups, both Theorems B and C can fail. We will exhibit a finitely-generated discrete group such that surjectivity fails in Theorem B for some increasing sequence of discrete -modules, and will then use this to show that surjectivity also fails in Theorem A for the inverse sequence of quotients , , and with target module equal to . We suspect that injectivity can also fail in both cases, but have not constructed examples to show this.
1.4 Topologies on cohomology groups and further comparison results
The remainder of our work concerns two other sets of questions about , which turn out to be closely intertwined.
First, each group inherits a quotient topology as the quotient of the group of measurable cocycles by that of coboundaries, and it is natural to ask when this topology is Hausdorff. This question has some intrinsic interest, but it also has consequences for the algebraic properties of the theory: an important part of Moo76(gr-cohomIII) is the development of a Lyndon-Hochschild-Serre spectral sequence for calculating the cohomology of a group extension, but this makes sense only when the topologies on the groups for the kernel of the extension are Hausdorff. The remarks following Lemma I.1.1 in Moo64(gr-cohomI-II) offer some further discussion of this issue, and it also arises in Chapter IX of Borel and Wallach BorWal00 (particularly Section IX.3) and Remark 12.0.6 in Monod Mon01 for related theories that are specific to Fréchet target modules.
Some special cases in which the quotient topology is Hausdorff are given in Moo64(gr-cohomI-II); Moo76(gr-cohomIII); Moo76(gr-cohomIV). The following result is rather more general than those.
Theorem D If is almost connected (that is, its identity component is co-compact) then the cohomology groups are Hausdorff in their quotient topologies in all of the following cases:
a Euclidean space, in which case each is also Euclidean in its quotient topology;
discrete, in which case each is discrete and countable;
a torus, in which case each is of the form
locally compact and locally contractible and with trivial -action, in which case is of the form
(Recall that an Abelian topological group is locally compact and locally contractible if and only if it is a Lie group, although we shall generally prefer the former description in the sequel.)
The second set of questions concerns the relation between and the other theories and .
Hofmann and Mostert HofMos73 show that for discrete target modules enjoys the same continuity properties as asserted by Theorems B and C, and more recently Flach has shown that also has these continuity properties (Proposition 8.1 in Fla08). Combined with Wigner’s, Lichtenbaum’s and Flach’s results that these theories coincide for Lie groups and discrete target modules (see Theorem 4 in Wig73, Section 2 of Lic09 and Proposition 5.2 in Fla08), this proves that all three theories coincide for compact base groups and discrete targets.
Using this special case, it is then possible to analyse more general locally compact second countable groups by applying the Lyndon-Hochschild-Serre spectral sequence to the presentation of such a group implied by the Gleason-Montgomery-Zippin Theorem on the resolution of Hilbert’s Fifth Problem. Since this spectral sequence makes the technical requirement that the cohomology groups of the kernel of a group extension be Hausdorff, Theorem D plays an important rôle in this extension of the comparison results between the various theories.
Our main new comparison result is the following.
Theorem E If is a locally compact group and is a -module which is either Fréchet or locally compact and locally contractible, then
In addition, if is Fréchet then this agrees with ; if is locally compact and locally contractible, then it agrees with , since such an lies in Segal’s category of modules; and if is discrete, then these also agree with .
We also offer a simple example in which both and are compact to show that does not always coincide with either or .
In comparing , , and , one should note that each has the virtue of being the unique solution to a universal problem — namely an effaceable cohomological functor on its category of definition. Indeed, for comparing just and one may choose the categories to be the same, and the difference appears only in what are the permitted ‘short exact sequences’ to which the functor must assign long exact sequences. (Hence this difference is very much in the spirit of ‘relative homological algebra’: see, for instance, Section VI.2 of Brown Bro82 for a more classical example.)
In the case of this category is and all short exact sequences as in (1) are allowed. This provides the natural setting for applications to functional analysis or representation theory, and so from the viewpoint of such applications the measurable-cochains theory is distinguished. It also has the virtue of having an equivalent definition without mention of cocycles in terms of equivalence classes of Yoneda-type multiple extensions of elements of . However, one disadvantage of is that computations of it are quite difficult in degrees greater than two. On the other hand , while it enjoys its universal property only in a more abstract category of semi-simplicial sheaves, is defined using a spectral sequence in sheaf cohomology that is already a powerful computational tool. Similarly , which has no obvious universality properties, can sometimes be computed very readily using tools from algebraic topology (as in HofMos73, for example). Hence Theorem E, which expands the known areas of agreement between , and , strengthens the usefulness of all three theories.
In fact, in the present paper we will already make use of this connexion in the proofs of some parts of Theorem D. This does not make the relationship between Theorem D and Theorem E circular: rather, we will first prove some special cases of Theorem E without using the LHS spectral sequence, then use these to obtain Theorem D, and then with this in hand return to more general cases of Theorem E.
1.5 Another application
Although this paper is purely about cohomology theories for topological groups, we note in passing that it was originally motivated by a very concrete application in ergodic theory, to questions concerning the structure of ‘characteristic factors’ for certain nonconventional ergodic averages. In the case recently treated in Aus--lindeppleasant1; Aus--lindeppleasant2, part of this structure could be characterized using a family of Borel -cocycles on a compact Abelian group taking values in the -module of all affine -valued maps on , endowed with the rotation action of . These -cocycles emerge in the description of some isometric circle extension of a measurable probability-preserving -action by rotations on , and more specifically as the obstructions to the factorizability of some -cocycle that actually corresponds to an extension of acting groups. It is shown in Aus--lindeppleasant2 that all examples of such systems can be obtained as inverse limits of finite-dimensional examples (in a certain natural sense), and these in turn can always be assembled by performing some standard manipulations on a special class of examples (joinings of partially invariant systems and two-step nilsystems). The continuity results of the present paper were at the heart of the reduction to finite-dimensional examples in that work, and so provided a crucial ingredient needed in the proof of the main structure theorem in Aus--lindeppleasant2. We will not give a more complete introduction to these questions here, but the whole story can be found in those papers.
Throughout this paper we will work with the presentation of the cohomology theory using the inhomogeneous bar resolution, and refer to Moo64(gr-cohomI-II); Moo76(gr-cohomIII); Moo76(gr-cohomIV) for the relation of this to other definitions.
Suppose that is a locally compact second countable group, and let denote a left-invariant Haar measure on , normalized to be a probability if is compact. These assumptions will now stand for the rest of this paper unless explicitly contradicted. A Polish Abelian -module is a triple in which is a Polish Abelian group with translation-invariant metric and is an action by continuous automorphisms. It is Fréchet if is also a separable and locally convex real topological vector space in its Polish topology, and it is Euclidean if in addition it is finite dimensional. It will later prove helpful to keep the metric and action explicit. Since is continuous, if is compact then by averaging the shifted metrics if necessary we may assume that is also -invariant.
If is a Polish -module and then we write for the -module of Haar-a.e. equivalence classes of Borel maps with the topology of convergence in probability (defined using the topology of ). Since we assume is second countable this topology on is also Polish. Beware that the notation ‘’ indicates ‘cochains’, as usual in group cohomology, and not ‘continuous functions’ — continuity will be marked by a subscript, as will be exhibited shortly.
When we interpret as , and when we interpret it as the trivial group . When necessary we will equip it with the diagonal action
We denote the identity in or in any by .
We also write , and define the coboundary maps by
(where this and all similar later equations are to be understood as holding Haar-almost-everywhere; issues surrounding this point are discussed carefully in Section 4 of Moo76(gr-cohomIII), and we will not dwell on them here). We define and . As usual, one verifies that , and so and we can define
these are the measurable cohomology groups for with coefficients in .
Exactly analogously, we define to consist of cochains that are continuous, and within it the subspaces and of continuous cocycles and coboundaries. The resulting cohomology groups
are the continuous cohomology groups for with coefficients in .
The various other cohomology theories for topological groups mentioned in the Introduction will be quickly recalled when we begin their analysis in Section LABEL:sec:other-theories.
If is a homomorphism of groups and then we write
so that . Clearly
so this lifting homomorphism has a quotient , referred to as the inflation homomorphism associated to .
In case is compact, a Borel map is -small for some if
(so this definition implicitly involves a choice of metric on ). For two such maps , we define
This is routinely verified to define a metric on which metrizes the topology of convergence in probability; when this gives the usual F-space structure on , which space is often denoted by in functional analysis. We also define the uniform metric on associated to by
of course for arbitrary Borel maps this may take the value .
We will sometimes use some standard analyst’s notation: the relation asserts that is bounded by the product of with some positive constant depending only on , and similarly a quantity is if it is bounded by the product of with some positive constant depending only on .
3 Improving the regularity of cocycles
3.1 Recap of dimension-shifting
Our later arguments will rely crucially on the procedure of dimension-shifting, which allows us to re-write one cohomology group as another of different degree (and with a different target module), and so by induction on degree gain access to algebraic properties that are manifest only in low degrees (usually degree one).
This possibility follows from the standard long exact sequence together with effacement. Concretely, we will use the vanishing result that for all and for any when is equipped with the diagonal action (see Theorem 4 of Moo76(gr-cohomIII)). Indeed, any -module embeds into as the closed submodule of constant maps; let be this embedding. As a result of the vanishing, constructing the long exact sequence from the presentation
collapses to give a sequence of switchback maps that are isomorphisms for all .
As is standard, the switchback map obtained above is implemented by a simple operator from into : if then we define by
Now a straightforward manipulation of the equation shows that is actually the constant-valued map . We record this here for convenience: for and we have from the definition (2)
Thus, the image of under the quotient defines a class in , and the usual diagram chase shows that it depends only on : this new class is the image of under the switchback isomorphism.
3.2 The regularizing argument: qualitative version
The measurable crossed homomorphisms that appear in degree- cohomology are automatically continuous, and the first benefit of dimension-shifting for our work is that it allows us to derive consequences of this in higher degrees. We will now prove a basic result giving a sense in which all cohomology classes can be represented by cocycles having some additional regularity, not possessed by arbitrary measurable cochains.
A Borel map from a locally compact space to a Polish space is locally totally bounded if for any compact the image is precompact in .
We will need the following classical result from the topology of metric spaces: it follows, for instance, from Proposition 18 in Section IX.2 of Bourbaki Bourb89.
If is an inclusion of Polish Abelian groups (so is closed in ) and is the quotient map, then for any compact subset there is a compact subset with . ∎
For any locally compact, second countable and Polish -module , any cohomology class in has a representative which is locally totally bounded.
This follows by a dimension-shifting induction. When it is immediate from the automatic continuity of -cocycles, so suppose now that and is a -cocycle.
By dimension-shifting there is a cochain with , so that the image of under the quotient onto is a -cocycle. Therefore by the inductive hypothesis it is equal to for some and some locally totally bounded cocycle .
By choosing a Borel partition of that is countable, locally finite and has each cell precompact, and then applying the preceding lemma on each cell, there are measurable lifts and of and so that is still locally totally bounded. It follows that
for some , and hence
Now, on the one hand, must takes values in the subgroup , and so is identified with an element of cohomologous to . On the other, for any compact one has the property that
so this is precompact as a subset of by the local total boundedness of . Since the topology of the closed subgroup agrees with the subspace topology, this implies that is locally totally bounded, so the induction continues. ∎
3.3 The regularizing argument: quantitative version
Proposition 3.3 is already enough to derive some consequences on the structure of the groups or the comparison with other cohomology theories, but the continuity results of Theorems B and C for discrete or toral targets are more delicate. These will rest on a quantitative analog of Proposition 3.3 which in some settings will give us an additional ability to prove the triviality of cocycles that are quantitatively ‘small enough’.
We will need the following elementary estimate relating the sizes of and , where is the explicit dimension-shifting operator introduced at the beginning of this section.
If is compact and is -small with , then is -small as a map from to the module : that is,
The above measure is simply
so since the map
preserves Haar measure this follows directly from Fubini’s Theorem. ∎
The next lemma is a quantitative analog of the standard result that crossed homomorphisms are always continuous.
Suppose that is compact, that is a Polish -module and that is a Borel crossed homomorphism. If is -small for some then .
Consider the crossed homomorphism equation
Let , so by assumption . Therefore for any , and so for any we can find so that also . Now since and are both within distance of and preserves , the above equation gives that is within distance of . ∎
There is a sequence of absolute constants for which the following holds. Suppose that is compact, that is a Polish -module, that and that is -small for some . Then for some that is -small and some with .
The construction of and the proof proceed by induction on .
Base clause: If we choose then in this case any is a crossed homomorphism and , so it must already satisfy by Lemma 3.5.
Recursion clause Suppose we know the assertion in all degrees up to some and for some , , …, , and we wish to prove it in degree . Given an -small , the dimension-shifting operator gives that is -small and is such that its image upon quotienting by is a member of . Of course is also -small, so applying the inductive hypothesis to we see that it is of the form for some -small and some for which
Making two applications of the measurable selection theorem, we may select some -small lifting and some lifting such that
and for these it follows that for some that takes values in the subgroup of constant maps.
Since , and are all -small as maps from into the module , it follows that is also -small for some . Provided we chose for some sufficiently small, this implies that , and hence that there is some subset of of measure at least for which the map is -small, and so takes values within of on a subset of of measure at least . However, since actually takes values in the subgroup of constant-valued maps, we may therefore identify it with a -small member of . Note that the assumption that was crucial here in justifying the implication that a constant-valued function is -small as a member of only if its constant value lies within of .
Finally applying the coboundary operator gives
where is -small, and if we set then
on the one hand we have
since for every the map is a sum of different values of , all of which satisfy such an essential supremum bound;
and on the other both and take values in the subgroup , and hence the same must be true of .
Therefore, once again making the proviso that be sufficiently small to guarantee that , we deduce that the constant values taken by must lie within of almost surely, and so the induction continues to . ∎
Remark A fairly crude check shows that the sequence is certainly small enough.
4 Fréchet modules and continuous cocycles
Theorem A follows quite quickly from Proposition 3.3.
Proof of Theorem A This is known in degree by the automatic continuity of crossed homomorphisms. We will show that for any and any Fréchet -module , any cohomology class in has a representative cocycle that is effaced by the inclusion
where the latter is given its compact-open topology. This shows that remains effaceable if it is restricted to the category of Fréchet -modules, since is still an object of this category (unlike , which is in general a non-locally-convex F-space). The continuous-cochains theory is also effaceable under the same inclusion, and it also has long exact sequences because quotients of Fréchet spaces admit continuous lifts by the Bartle-Graves-Michael selection theorems (see, for instance, Section 1.3 of Benyamini and Lindenstrauss