We recall Brouwer’s famous fixed point theorem:
This theorem has many proofs, most of which revolve (either explicitly or implicitly) around the notion of the degree of a continuous map of the unit sphere to itself, and more precisely around the stability of degree with respect to homotopy. (Indeed, one can view the Brouwer fixed point theorem as an assertion that some non-trivial degree-like invariant must exist, or more abstractly that the homotopy group is non-trivial.)
One of the many applications of this result is to prove Brouwer’s invariance of domain theorem:
This theorem in turn has an important corollary:
This corollary is intuitively obvious, but note that topological intuition is not always rigorous. For instance, it is intuitively plausible that there should be no continuous surjection from to for , but such surjections always exist, thanks to variants of the Peano curve construction.
Theorem 2 or Corollary 3 can be proven by simple ad hoc means for small values of or (for instance, by noting that removing a point from will disconnect when , but not for ), but I do not know of any proof of these results in general dimension that does not require algebraic topology machinery that is at least as sophisticated as the Brouwer fixed point theorem. (Lebesgue, for instance, famously failed to establish the above corollary rigorously, although he did end up discovering the important concept of Lebesgue covering dimension as a result of his efforts.)
Nowadays, the invariance of domain theorem is usually proven using the machinery of singular homology. In this post I would like to record a short proof of Theorem 2 using Theorem 1 that I discovered in a paper of Kulpa, which avoids any use of algebraic topology tools beyond the fixed point theorem, though it is more ad hoc in its approach than the systematic singular homology approach.
Remark 1 A heuristic explanation as to why the Brouwer fixed point theorem is more or less a necessary ingredient in the proof of the invariance of domain theorem is that a counterexample to the former result could conceivably be used to create a counterexample to the latter one. Indeed, if the Brouwer fixed point theorem failed, then (as is well known) one would be able to find a continuous function that was the identity on (indeed, one could take to be the first point in which the ray from through hits ). If one then considered the function defined by , then this would be a continuous function which avoids the interior of , but which maps the origin to a point on the sphere (and maps to the dilate ). This could conceivably be a counterexample to Theorem 2, except that is not necessarily injective. I do not know if there is a more rigorous way to formulate this connection.
The reason I was looking for a proof of the invariance of domain theorem was that it comes up in the very last stage of the solution to Hilbert’s fifth problem, namely to establish the following fact:
Recall that a locally Euclidean group is a topological group which is locally homeomorphic to an open subset of a Euclidean space , i.e. it is a continuous manifold. Note in contrast that a Lie group is a topological group which is locally diffeomorphic to an open subset of , it is a smooth manifold. Thus, Hilbert’s fifth problem is a manifestation of the “rigidity” of algebraic structure (in this case, group structure), which turns weak regularity (continuity) into strong regularity (smoothness).
It is plausible that something like Corollary 3 would need to be invoked in order to solve Hilbert’s fifth problem. After all, if Euclidean spaces , of different dimension were homeomorphic to each other, then the property of being locally Euclidean loses a lot of meaning, and would thus not be a particularly powerful hypothesis. Note also that it is clear that two Lie groups can only be isomorphic if they have the same dimension, so in view of Theorem 4, it becomes plausible that two Euclidean spaces can only be homeomorphic if they have the same dimension, although I do not know of a way to rigorously deduce this claim from Theorem 4.
Interestingly, Corollary 3 is the only place where algebraic topology enters into the solution of Hilbert’s fifth problem (although its cousin, point-set topology, is used all over the place). There are results closely related to Theorem 4, such as the Gleason-Yamabe theorem mentioned in a recent post, which do not use the notion of being locally Euclidean, and do not require algebraic topological methods in their proof. Indeed, one can deduce Theorem 4 from the Gleason-Yamabe theorem and invariance of domain; we sketch a proof of this (following Montgomery and Zippin) below the fold.
— 1. Invariance of domain —
Now we prove Theorem 2. By rescaling and translation invariance, it will suffice to show the following claim:
Let be as in Theorem 5. The map is a continuous bijection between compact Hausdorff spaces and is thus a homeomorphism. In particular, the inverse map is continuous. Using the Tietze extension theorem, we can find a continuous function that extends .
The function has a zero on , namely at . We can use the Brouwer fixed point theorem to show that this zero is stable:
Proof: Apply Theorem 1 to the function
We turn to the details. Let be a small number. By continuity of , we see (if is chosen small enough) that we have whenever and .
Let denote the set , where
Note that is continuous and well-defined since avoids zero. Informally, is a perturbation of caused by pushing out a small distance away from the origin (and hence also away from ), with being the “pushing” map.
By construction, is non-zero on ; since is compact, is bounded from below on by some . By shrinking if necessary we may assume that .
By the Weierstrass approximation theorem, we can find a polynomial such that
for all ; in particular, does not vanish on . At present, it is possible that vanishes on . But as is smooth and has measure zero, also has measure zero; so by shifting by a small generic constant we may assume without loss of generality that also does not vanish on . (If one wishes, one can use an algebraic geometry argument here instead of a measure-theoretic one, noting that lies in an algebraic hypersurface and can thus be generically avoided by perturbation. A purely topological way to avoid zeroes in is also given in Kulpa’s paper.)
Now consider the function defined by
and hence by (3) and the triangle inequality
Thus in all cases we have
for all . But this, combined with the non-vanishing nature of , contradicts Lemma 6.
— 2. Hilbert’s fifth problem —
We now sketch how invariance of domain can be used to establish the solution to Hilbert’s fifth problem (as formulated in Theorem 4). Our main tool is the Gleason-Yamabe theorem, in the form stated in Theorem 4 of this previous post:
Theorem 7 (Gleason-Yamabe theorem) Every locally compact Hausdorff group has an open subgroup that is the projective limit of Lie groups.
A locally Euclidean group is of course locally compact Hausdorff; it is also first countable, and hence (by the Birkhoff-Kakutani theorem, discussed in this post) is also metrisable. Because of this, it is not difficult to show that the open subgroup given by the Gleason-Yamabe theorem in this case can be obtained as a projective limit
of a countable sequence of Lie groups, with continuous homomorphisms from to for each .
Using Cartan’s theorem to obtain the smoothness of outer automorphisms on Lie groups as in previous posts, we see that to show that a topological group is a Lie group, it suffices to locate an open subgroup that is a Lie group. In view of these reductions, it suffices to show
Proposition 8 Let be a locally Euclidean group that is the inverse limit
of Lie groups. Then is itself isomorphic to a Lie group.
We now prove the proposition. We first observe by shrinking the if necessary (and using Cartan’s theorem that locally compact subgroups of a Lie group are still Lie) we may assume that all the projection maps in the inverse limit are surjective; indeed, by local compactness we may take where are a sequence of compact normal subgroups converging to the identity.
As continuous homomorphisms of Lie groups are automatically smooth (as proven in this previous post), the projection maps from to induce an associated projection map from to at the Lie algebra level. As the former maps are surjective, the latter maps are also. In particular, the dimensions of the finite-dimensional Lie algebras are non-decreasing in .
Another consequence of surjectivity is that every tangent vector in can be lifted up to . Continuing this process and passing to the inverse limit, it is possible to show that every one-parameter subgroup in one of the factor groups can be lifted up (locally, at least) to a one-parameter subgroup of . In fact, one can show that a neighbourhood of the origin in can be lifted up into , to provide a continuous injection of a non-empty open subset of that Lie algebra into .
Now we crucially use the invariance of domain (Corollary 3) to conclude that the dimensions of the Lie algebras must be bounded in (indeed, they cannot exceed the dimension of the locally Euclidean group ). As these dimensions are non-decreasing, they must therefore be eventually constant; by discarding finitely many of the factor groups, we may thus assume that all the have the same dimension. In particular, the kernels of the projection maps from to are zero-dimensional Lie groups and are thus discrete. As , this means that the are also discrete, thus is an open subgroup of , and so (by compactness of ) has finite index. As the converge to the trivial group as , we conclude that each of the are profinite, and in particular are totally disconnected. Thus, for any , we have a short exact sequence
describing as the extension of a Lie group by a totally disconnected compact group . Furthermore, we can lift a small neighbourhood of the identity of to in a continuously injective manner. This neighbourhood is connected, and acts on the normal subgroup by conjugation. But is totally disconnected, and so all orbits of this action must be points; thus, the lift of this neighbourhood commutes with . As a consequence, one can show that has the local topological (and group) structure of the direct product . But is also locally Euclidean; as is totally disconnected, these two statements are only compatible if is discrete. Thus the Lie group is an open subgroup of , and we are done.
Remark 2 The above argument shows that any metrisable group which is “finite-dimensional” in the sense that it does not contain continuous injective images of non-trivial open sets of Euclidean spaces of arbitrarily large dimension, has the local structure (as a topological space) of the product of a Euclidean space and a totally disconnected space. Of course, direct products of Lie groups and totally disconnected groups provide one such example of this claim. A more non-trivial example is given by the solenoid groups, a simple example of which is the (additive) group
where is the diagonally embedded copy of the integers. This is a compact Hausdorff group that has a short exact sequence
and has the local structure of , although does not embed into this group and so this is not a direct or semi-direct product. (Topologically, it can be viewed as the set after identifying with for every -adic integer .) It has topological dimension one, in the sense that it contains continuous injective images of non-empty open subsets of if and only if (Exercise!). It is also the projective limit of the Lie groups
each of which is isomorphic to the circle , but in a manner which becomes increasingly “twisted” as increases (which helps explain the terminology “solenoid”). It is an instructive exercise to verify that is connected, but not path connected or locally connected. Thus we see that connected locally compact groups can contain some significant non-Lie behaviour, even in the abelian setting.
Remark 3 Of course, it is possible for locally compact groups to be infinite-dimensional; a simple example is the infinite-dimensional torus , which is compact, abelian, metrisable, and locally connected, but infinite dimensional. (It will still be an inverse limit of Lie groups, though.)
Remark 4 The above analysis also gives another purely topological characterisation of Lie groups; a topological group is Lie if and only if it is locally compact, Hausdorff, first-countable, locally connected, and finite-dimensional. It is interesting to note that this characterisation barely uses the real numbers , which are of course fundamental in defining the smooth structure of a Lie group; the only remaining reference to comes through the notion of finite dimensionality. It is also possible, using dimension theory, to obtain alternate characterisations of finite dimensionality (e.g. finite Lebesgue covering dimension) that avoid explicit mention of the real line, thus capturing the concept of a Lie group using only the concepts of point-set topology (and the concept of a group, of course).