In the last few months, I have been working my way through the theory behind the solution to Hilbert’s fifth problem, as I (together with Emmanuel Breuillard, Ben Green, and Tom Sanders) have found this theory to be useful in obtaining noncommutative inverse sumset theorems in arbitrary groups; I hope to be able to report on this connection at some later point on this blog. Among other things, this theory achieves the remarkable feat of creating a smooth Lie group structure out of what is ostensibly a much weaker structure, namely the structure of a locally compact group. The ability of algebraic structure (in this case, group structure) to upgrade weak regularity (in this case, continuous structure) to strong regularity (in this case, smooth and even analytic structure) seems to be a recurring theme in mathematics, and an important part of what I like to call the “dichotomy between structure and randomness”.
The theory of Hilbert’s fifth problem sprawls across many subfields of mathematics: Lie theory, representation theory, group theory, nonabelian Fourier analysis, point-set topology, and even a little bit of group cohomology. The latter aspect of this theory is what I want to focus on today. The general question that comes into play here is the extension problem: given two (topological or Lie) groups and , what is the structure of the possible groups that are formed by extending by . In other words, given a short exact sequence
to what extent is the structure of determined by that of and ?
As an example of why understanding the extension problem would help in structural theory, let us consider the task of classifying the structure of a Lie group . Firstly, we factor out the connected component of the identity as
as Lie groups are locally connected, is discrete. Thus, to understand general Lie groups, it suffices to understand the extensions of discrete groups by connected Lie groups.
Next, to study a connected Lie group , we can consider the conjugation action on the Lie algebra , which gives the adjoint representation . The kernel of this representation consists of all the group elements that commute with all elements of the Lie algebra, and thus (by connectedness) is the center of . The adjoint representation is then faithful on the quotient . The short exact sequence
then describes as a central extension (by the abelian Lie group ) of , which is a connected Lie group with a faithful finite-dimensional linear representation.
This suggests a route to Hilbert’s fifth problem, at least in the case of connected groups . Let be a connected locally compact group that we hope to demonstrate is isomorphic to a Lie group. As discussed in a previous post, we first form the space of one-parameter subgroups of (which should, eventually, become the Lie algebra of ). Hopefully, has the structure of a vector space. The group acts on by conjugation; this action should be both continuous and linear, giving an “adjoint representation” . The kernel of this representation should then be the center of . The quotient is locally compact and has a faithful linear representation, and is thus a Lie group by von Neumann’s version of Cartan’s theorem (discussed in this previous post). The group is locally compact abelian, and so it should be a relatively easy task to establish that it is also a Lie group. To finish the job, one needs the following result:
This result can be obtained by combining a result of Kuranishi with a result of Gleason; I am recording this argument below the fold. The point here is that while is initially only a topological group, the smooth structures of and can be combined (after a little bit of cohomology) to create the smooth structure on required to upgrade from a topological group to a Lie group. One of the main ideas here is to improve the behaviour of a cocycle by averaging it; this basic trick is helpful elsewhere in the theory, resolving a number of cohomological issues in topological group theory. The result can be generalised to show in fact that arbitrary (topological) extensions of Lie groups by Lie groups remain Lie; this was shown by Gleason. However, the above special case of this result is already sufficient (in conjunction with the rest of the theory, of course) to resolve Hilbert’s fifth problem.
Remark 1 We have shown in the above discussion that every connected Lie group is a central extension (by an abelian Lie group) of a Lie group with a faithful continuous linear representation. It is natural to ask whether this central extension is necessary. Unfortunately, not every connected Lie group admits a faithful continuous linear representation. An example (due to Birkhoff) is the Heisenberg-Weyl group
Indeed, if we consider the group elements
for some prime , then one easily verifies that has order and is central, and that is conjugate to . If we have a faithful linear representation of , then must have at least one eigenvalue that is a primitive root of unity. If is the eigenspace associated to , then must preserve , and be conjugate to on this space. This forces to have at least distinct eigenvalues on , and hence (and thus ) must have dimension at least . Letting we obtain a contradiction. (On the other hand, is certainly isomorphic to the extension of the linear group by the abelian group .)
— 1. A little group cohomology —
Let us first ignore the topological or Lie structure, consider the (central) extension problem for discrete groups only. Thus, let us suppose we have a (discrete) group which is a central extension of a group by a group . We view as a central subgroup of (which we write additively to emphasise its abelian nature), and use to denote the projection map. If and , we write for to emphasise the central nature of .
It may help to view as a principal -bundle over , with being thought of as the “vertical” component of and as the “horizontal” component. Thus is the union of “vertical fibres” , indexed by the horizontal group , each of which is a coset (or a torsor) of the vertical group .
As central extensions are not unique, we will need to specify some additional data beyond and . One way to view this data is to specify a section of the extension, that is to say a map that is a right-inverse for the projection map , thus selects one element from each fibre of the projection map (which is also a coset of ). Such a section can be always chosen using the axiom of choice, though of course there is no guarantee of any measurability, continuity, or smoothness properties of such a section if the groups involved have the relevant measure-theoretic, topological, or smooth structure.
Note that we do not necessarily require the section to map the goup identity of to the group identity of , though in practice it is usually not difficult to impose this constraint if desired.
The group can then be described, as a set, as the disjoint union of its fibres for :
Thus each element can be uniquely expressed as for some and , and can thus be viewed as a system of coordinates of (identifying it as a set with ). Now we turn to the group operations and . If , then the product must lie in the fibre of , and so we have
for all ; we refer to functions that obey this equation as cocycles (the reason for this terminology being explained in this previous blog post). The space of all such cocyles is denoted (or , if we wish to emphasise that we are working for now in the discrete category, as opposed to the measurable, topological, or smooth category); this is an abelian group with respect to pointwise addition. Using (1), we also have a description of the group identity of in coordinates:
Thus we see that the cocycle , together with the group structures on and , capture all the group-theoretic structure of . Conversely, given any cocycle , one can place a group structure on the set by declaring the multiplication law as
the identity element as , and the inversion law as
Exercise 1 Using the cocycle equation (2), show that the above operations do indeed yield group structure on (i.e. the group axioms are obeyed). If arises as the cocycle associated to a section of a group extension , we then see from (1), (3), (4) that this group structure we have just placed on is isomorphic to that on .
Thus we see that once we select a section, we can describe a central extension of by (up to group isomorphism) as a cocycle, and conversely every cocycle arises in this manner. However, we have some freedom in deciding how to select this section. Given one section , any other section takes the form
for some function ; conversely, every such function can be used to shift a section to a new section . We refer to such functions as gauge functions. The cocycles associated to are related by the gauge transformation
where is the function
We refer to functions of the form as coboundaries, and denote the space of all such coboundaries as (or , if we want to emphasise the discrete nature of these coboundaries). One easily verifies that all coboundaries are cocycles, and so is a subgroup of . We then define the second group cohomology (or ) to be the quotient group
and refer to elements of as cohomology classes. (There are higher order group cohomologies, which also have some relevance for the extension problem, but will not be needed here; see this previous post for further discussion. The first group cohomology is just the space of homomorphisms from to ; again, this has some relevance for the extension problem but will not be needed here.)
We call two cocycles cohomologous if they differ by a coboundary (i.e. they lie in the same cohomology class). Thus we see that different sections of a single central group extension provide cohomologous cocycles. Conversely, if two cocycles are cohomologous, then the group structures on given by these cocycles are easily seen to be isomorphic; furthermore, if we restrict group isomorphism to fix each fibre of , this is the only way in which group structures generated by such cocycles are isomorphic. Thus, we see that up to group isomorphism, central group extensions are described by cohomology classes.
Remark 2 All constant cocycles are coboundaries, and so if desired one can always normalise a cocycle (up to coboundaries) so that . This can lead to some minor simplifications to some of the cocycle formulae (such as (3), (4)).
The trivial cohomology class of course contains the trivial cocycle, which in turn generates the direct product . Non-trivial cohomology classes generate “skew” products that will not be isomorphic to the direct product (at least if we insist on fixing each fibre). In particular, we see that if the second group cohomology is trivial, then the only central extensions of by are (up to isomorphism) the direct product extension.
In general, we do not expect the cohomology group to be trivial: not every cocycle is a coboundary. A simple example is provided by viewing the integer group as a central extension of a cyclic group by the subgroup using the short exact sequence
Clearly is not isomorphic to (the former group is torsion-free, while the latter group is not), and so is non-trivial. If we use the section that maps to for , then the associated cocycle is the familiar “carry bit” one learns about in primary school, which equals when and , but vanishes for other .
However, when the “vertical group” is sufficiently “Euclidean” in nature, one can start deploying the method of averaging to improve the behaviour of cocycles. Here is a simple example of the averaging method in action:
Proof: We need to show that every cocycle is a coboundary. Accordingly, let be a cocycle, thus
for all . We sum this equation over all , and then divide by the cardinality of ; note that this averaging operation relies on the finite nature of and the vector space nature of . We obtain as a consequence
is the averaging of in the second variable. We thus have , and so is a coboundary as desired.
It is instructive to compare this argument against the non-triviality of mentioned earlier. While is still finite, the problem here is that is not a vector space and so one cannot divide by to “straighten” the cocycle. However, the averaging argument can still achieve some simplification to such cocycles:
Exercise 2 Let be a finite group, and let is a cocycle. Show that is cohomologous to a cocycle taking values in . (Hint: average the cocycle as before using the reals , then round to the nearest integer.)
Now we turn from discrete group theory to topological group theory. Now , , are required to be topological groups rather than discrete groups, with the projection map continuous. (In particular, if is Hausdorff, then must be closed.) In this case, it is no longer the case that every (discrete) cocycle gives rise to a topological group, because the group structure on given by need not be continuous with respect to the product topology of . However, if the cocycle is continuous, then it is clear that the group operations will be continuous, and so we will in fact generate a topological group. Conversely, if we have a section which is continuous, then it is not difficult to verify that the cocycle generated is also continuous. This leads to a slightly different group cohomology, using the space of continuous cocycles, and also the space of coboundaries arising from continuous gauge functions .
However, such “global” cohomology contains some nontrivial global topological obstructions that limit its usefulness for the extension problem. One particularly fundamental such obstruction is that one does not expect global continuous sections of a group extension to exist in general, unless or have a particularly simple topology (e.g. if they are contractible or simply connected). For instance, with the short exact sequence
there is no way to continuously lift the horizontal group back up to the real line , due to the presence of nontrivial monodromy. While these global obstructions are quite interesting from an algebraic topology perspective, they are not of central importance in the theory surrounding Hilbert’s fifth problem, which is instead more concerned with the local topological and Lie structure of groups.
Because of this, we shall work with local sections and cocycles instead of global ones. A local section in a topological group extension
is a continuous map from an open neighbourhood of the identity in to which is a right inverse of on , thus for all . (It is not yet obvious why local sections exist at all – there may still be local obstructions to trivialising the fibre bundle – but certainly this task should be easier than that of locating global continuous sections.)
Similarly, a local cocycle is a continuous map defined on some open neighbourhood of the identity in that obeys the cocycle equation (2) whenever are such that . We consider two local sections , to be locally identical if there exists a neighbourhood of the identity contained in both and such that and agree on . Similarly, we consider two local cocycles and to be locally identical if there is a neighbourhood of the identity in such that and agree on . We let be the space of local cocycles, modulo local identity; this is easily seen to be an abelian group.
One easily verifies that every local section induces a local cocycle (perhaps after shrinking the open neighbourhood slightly), which is well-defined up to local identity. Conversely, the computations used to show Exercise 1 show that every local cocycle creates a local group structure on .
If is an open neighbourhood the identity of , is a continuous gauge function, and is a smaller neighbourhood such that , we can define the local coboundary by the usual formula
This is well-defined up to local identity. We let be the space of local coboundaries, modulo local identity; this is a subgroup of . Thus, one can form the local topological group cohomology . We say that two local cocycles are locally cohomologous if they differ by a local coboundary.
The relevance of local cohomology to the Lie group extension problem can be seen from the following lemma.
Lemma 3 Let be a central (topological) group extension of a Lie group by a Lie group . If there is a local section whose associated local cocycle is locally cohomologous to a smooth local cocycle , then is also isomorphic to a Lie group.
Note that the notion of smoothness of a (local) cocycle makes sense, because there are smooth structures in place for both (and hence ) and . In contrast, one cannot initially talk about a smooth (local) section , because initially only has a topological structure and not a smooth one.
By hypothesis, we locally have for some continuous gauge function . Rotating by , we thus obtain another local section whose associated local cocycle is smooth. We use this section to identify as a topological space with for some sufficiently small neighbourhood of the origin. We can then give a smooth structure induced from the product smooth structure on . By (1), (4) (and (3)), we see that the group operations are smooth on a neighbourhood of the identity in .
To finish the job and give the structure of a Lie group, we need to extend this smooth structure to the rest of in a manner which preserves the smooth nature of the group operations. Firstly, by using the local coordinates of we see that is locally connected. Thus, if is the connected component of the identity, then the quotient group .
Now we use the same argument that we gave in the post on von Neumann’s theorem. Firstly, we extend the smooth structure on a neighbourhood of the identity on to the rest of by (say) left-invariance; it is easy to see that these smooth coordinate patches are compatible with each other. A continuity argument then shows that the group operations are smooth on . Each element of acts via conjugation by a continuous homomorphism on the Lie group ; by applying Cartan’s theorem to the graph of this homomorphism as in the preceding post, we see that that such homomorphisms are smooth. From this we can then conclude that the group operations are smooth on all of .
— 2. Proof of theorem —
Lemma 3 suggests a strategy to prove Theorem 1: first, obtain a local section of , extract the associated local cocycle, then “straighten” it to a smooth cocycle. This will indeed be how we proceed. In obtaining the local section and in smoothing the cocycle we will take advantage of the averaging argument, adapted to the Lie algebra setting.
We begin by constructing the section, following this paper of Gleason. The idea is to use the zero set of a certain continuous function from into a finite-dimensional vector space . We will use topological arguments to force this function to have a zero at every fibre, and will give some “vertical non-degeneracy” to prevent it from having more than one zero on any given fibre, at least locally.
Let’s see how this works in the special case when the abelian Lie group is compact, so that (by the Peter-Weyl theorem, as discussed in this post) it can be modeled as a closed subgroup of a unitary group , thus we have an injective continuous homomorphism . We view as a subset of the vector space of complex matrices.
The group is locally compact and also Hausdorff (being the extension of one Hausdorff group by another). Thus, we may then apply the (LCH version of the) Tietze extension theorem (discussed on this blog here), and extend the map to a continuous (and compactly supported) function defined on all of .
for all , the extension need not obey any analogous symmetry in general. However, this can be rectified by an averaging argument. Define the function by the formula
where is normalised Haar measure on . Then is still continuous and still extends ; but, unlike , it now obeys the equivariance property
for and , as can easily be seen from (5).
The function is a better extension than , but suffers from another defect; while took values in the closed group , takes values in (for instance, is going to be compactly supported). There may well be global topological reasons that prevent from taking values in at all points, but as long as we are willing to work locally, we can repair this issue as follows. Using the inverse function theorem, one can find a manifold going through the identity matrix , which is transverse to in the sense that every matrix that is sufficiently close to the identity can be uniquely (and continuously) decomposed as , where and . (Indeed, to build , one can just use a complementary subspace to the tangent space (or Lie algebra) of at the identity.). For sufficiently close to the identity , we may thus factor
where and . By construction, will be continuous and take values in near , and equal on near ; it will also obey the equivariance
for and that are sufficiently close to the identity. From this, we see that for each that is sufficiently close to the identity, there exists a unique solution in to the equation that is also close to the identity; from the continuity of we see that is continuous near the identity and is thus a local section of as required.
Now we argue in the general case, in which the abelian Lie group need not be compact. If is connected, then it is isomorphic to a Euclidean space quotiented by a discrete subgroup. In particular, it can be quotiented down to a torus in a manner which is a local homeomorphism near the origin. Embedding that torus in a unitary group , we obtain an continuous homomorphism which is still locally injective and has compact image. One can then repeat the previous arguments to obtain a local section; we omit the details.
Finally, if is not connected, we can work with the connected component of the identity, which is locally the same as , and repeat the previous argument again to obtain a local section.
Using this local section, we obtain a local cocycle ; shifting by a constant, we may assume that . Near the identity, the abelian Lie group can be locally identified with a Euclidean space , so (shrinking if necessary) induces a -valued local cocycle .
We will smooth this cocycle by an averaging argument akin to the one used in Lemma 2. Instead of a discrete averaging, though, we will now use a non-trivial left-invariant Haar measure on (the existence of which is guaranteed by a classical theorem of Weil). As we only have a local cocycle, though, we will need to only average using a smooth, compactly supported function supported on a small neighbourhood of (small enough that ), normalised so that . For any , we have the cocycle equation
we average this against and conclude that
for , where is the function
and is the function
The function is continuous, and so is locally cohomologous to . Using the left-invariance of , we can rewrite as
from which it becomes clear that is smooth in the variable. A similar averaging argument (now against a bump function on a right-invariant measure) then shows that is in turn locally cohomologous to another local cocycle which is now smooth in both the and variables. Pulling back to , we conclude that is locally cohomologous to a smooth local cocycle, and Theorem 1 now follows from Lemma 3.