Poincaré’s stated motivations for topology

Abstract

It is well known that one of Poincaré’s most important contributions to mathematics is the creation of algebraic topology. In this paper, we examine carefully the stated motivations of Poincaré and potential applications he had in mind for developing topology. Besides being an interesting historical problem, this study will also shed some light on the broad interest of Poincaré in mathematics in a concrete way.

Appendix on finite and discrete subgroups of Lie groups

As mentioned before, the classification of finite subgroups of $GL(n,{\mathbb {C}})$ and other Lie groups and more generally of discrete subgroups of Lie groups was some of the motivations for Poincaré to develop algebraic topology. We add some comments to supplement our earlier explanations in Sect. 3.

It seems that a direct generalization of Klein’s method to determine finite subgroups of $GL(2, {\mathbb {C}})$ will not work for $n\ge 3$. There are several coincidences in the case $n=2$ which are used crucially. The symmetric space for $SL(2, {\mathbb {C}})$ is the three-dimensional real hyperbolic space ${\mathbb {H}}^3$, which can be realized as the unit ball in ${\mathbb {R}}^3$, and the unit sphere $S^2$ is the boundary of ${\mathbb {H}}^3$ and is also the flag variety of $SL(2, {\mathbb {C}})$. Furthermore, the isometry group of the unit sphere is SO(3), and SU(2) is a 2-to-1 cover of SO(3). They allow one to show that any finite subgroup of $GL(2, {\mathbb {C}})$ is conjugate to a subgroup of SU(2) and hence is reduced to finite subgroups of the rotation group SO(3). This was an essential part of the argument of Klein. All these fail when $n>3$. In this case, the symmetric space associated with $SL(n, {\mathbb {C}})$ is $SL(n, {\mathbb {C}})/SU(n)$ and is of higher rank, and no flag variety can be the full boundary of this symmetric space. Hence, there is no natural replacement for the analogue of SO(3).

In Jordan (1880), he used induction to prove that in the general case $GL(n, {\mathbb {C}})$, there are only finitely many types of finite subgroups. When $n=3$, he classified these finite subgroups by detailed computation through groups of permutation. There is no topology used in the proof of Jordan either. We note that Klein used geometry, but not topology, to prove his results. But geometry and topology probably went together in the mind of Poincaré.

It seems that Poincaré wanted to avoid the algebraic approach of Jordan and to use geometric and topological arguments instead. As explained above, a key step in Klein’s argument is to show that any finite subgroup of $SL(2, {\mathbb {C}})$ is conjugate to a subgroup of SU(2), which is a maximal compact subgroup of $SL(2, {\mathbb {C}})$. In fact, this follows from a general result in the theory of semisimple Lie groups: For any noncompact semisimple Lie group G, every two maximal compact subgroups of G are conjugate, and hence, every finite subgroup of G is conjugate to a finite subgroup of any fixed maximal compact subgroup. Consequently, every finite subgroup of $GL(n, {\mathbb {C}})$ is conjugate to a finite subgroup of SU(n).

The above general result was proved by E. Cartan much later than the works of Poincaré on topology. On the other hand, in this case of $G=SL(n, {\mathbb {C}})$, it can be given a direct proof. Let $Q_0(z, z)=\sum _{i=1}^n z_i \overline{z_i}$ be the standard Hermitian form on ${\mathbb {C}}^n$. For any finite subgroup $F\subset SL(n, {\mathbb {C}})$, consider the Hermitian form

$$\begin{aligned} Q(z, z)=\sum _{f\in F} Q_0( f z, f z). \end{aligned}$$

It is clearly positive definite and invariant under F. We can scale it to have determinant one. Then, Q will be conjugate to $Q_0$. Since the stabilizer in $SL(n, {\mathbb {C}})$ of $Q_0$ is equal to SU(n), it follows that the finite subgroup F is conjugate to a finite subgroup of SU(n).

To finish the rest of the argument of Klein, we need a classification of finite subgroups of SU(n). Though SU(n) is not related to SO(m) for some m by a finite-to-one map for $n\ge 3$ as in the case of $n=2$, SU(n) is contained in SO(2n) via the identification ${\mathbb {C}}={\mathbb {R}}^2$. On the other hand, the analogue of Jordan’s result on the classification of finite rotation subgroups of ${\mathbb {R}}^3$, i.e., finite subgroups of SO(3) (Jordan 1868, 1869), is not known for higher dimensions. For some discussion of finite subgroups of SO(4), see the book Conway and Smith (2003). Not much is known beyond that dimension.

It is not clear how the Euler characteristic can be used to study finite subgroups of $SL(n, {\mathbb {C}})$ and $GL(n, {\mathbb {C}})$, which was mentioned by Poincaré.

Remark 5.1

While reading the above comments in an earlier version of this paper, Gray raised a very interesting question: What did Bieberbach do when he solved this problem?

As it is known, Bieberbach solved in 1911 the 18th problem of Hilbert which asks:

“Is there in n-dimensional euclidean space also only a finite number of essentially different kinds of groups of motions with a fundamental region?”

In the above statement of Hilbert, he required the fundamental region to be compact, and such discrete groups of motions are usually called crystallographic groups. Bieberbach’s solution is summarized in Milnor (1976), 492 as follows:

• 1. Every crystallographic group $\Gamma$ in dimension n contains a free abelian subgroup T of rank n consisting of translations, which is necessarily of finite index in $\Gamma$. If $T\cong {\mathbb {Z}}^n$ is a maximal such subgroup of $\Gamma$, then $\Gamma$ admits an exact sequence

  $$\begin{aligned} 1 \rightarrow {\mathbb {Z}}^n\rightarrow \Gamma \rightarrow \Phi \rightarrow 1, \end{aligned}$$

  where $\Phi$ is a finite group which can be realized as a subgroup of $GL(n, {\mathbb {Z}})$. Note that $\Gamma$ is the semidirect product of $\Phi$ and ${\mathbb {Z}}^n$, with ${\mathbb {Z}}^n$ being the normal subgroup, and hence, $\Phi$ acts on ${\mathbb {Z}}^n$ and is contained in $GL(n, {\mathbb {Z}})$.

• 2. There are only finitely many isomorphism classes of such extensions $\Gamma$ of ${\mathbb {Z}}^n$ by finite subgroups $\Phi$ of $GL(n, {\mathbb {Z}})$.

For the second statement, Bieberbach used a result of Minkowski which states that there is a uniform upper bound depending only on n on the order of finite subgroups of $GL(n, {\mathbb {Z}})$ and proved that each finite group can only be realized in $GL(n, {\mathbb {Z}})$ in finitely many different embeddings. In all these arguments, algebraic methods were used.

In this problem, finite subgroups of $GL(n, {\mathbb {C}})$ are restricted to $GL(n, {\mathbb {Z}})$ in view of their action on the lattice ${\mathbb {Z}}^n$. Consequently, it becomes an easier problem than the problem on finite subgroups of $GL(n, {\mathbb {C}})$.

Maybe one geometric way to reprove a sub-case of Jordan’s result on the finiteness of types of finite subgroups is the following result: for any arithmetic subgroup, or more generally, a lattice$\Gamma$of $GL(n, {\mathbb {C}})$or of any other semisimple Lie group G, there are only finitely many conjugacy classes of finite subgroups in $\Gamma$.

Since $GL(n, {\mathbb {Z}})$ is a typical arithmetic subgroup, this result is a generalization of Minkowski’s result mentioned in the above remark.

This follows from the reduction theory of arithmetic groups acting on symmetric spaces of G. See the books Raghunathan (1972) and Borel (1969), for example. In some sense, it is related to the geometric method of Klein mentioned above since the action of the arithmetic subgroup on the symmetric spaces, which are complete Riemannian manifold of nonpositive curvature, is used crucially in the above proof.

In this context, finite groups and the Euler characteristic can appear in the Euler characteristic for locally symmetric spaces which contain orbifold singularities, or for discrete subgroups which are not torsion-free. See, for example, the paper Brown (1974) and its references.

Remark 5.2

It is perhaps interesting to mention that the standard books on infinite discrete subgroups, especially on lattices, of semisimple Lie groups such as Raghunathan (1972), Margulis (1991) have not mentioned that the problem to study such discrete subgroups was raised by Poincaré. On the other hand, it was mentioned explicitly by Jacques Tits in his description of Margulis’ Fields medal works (Tits 1980, ICM proceedings 1978, 58), which also showed the fruitfulness of this problem:

Already Poincaré wondered about the possibility of describing all discrete subgroups of finite covolume in a Lie group G. The profusion of such subgroups in $G = PSL_2({\mathbf {R}})$ makes one at first doubt of any such possibility. However, $PSL_2({\mathbf {R}})$ was for a long time the only simple Lie group which was known to contain non-arithmetic discrete subgroups of finite covolume, and further examples discovered in 1965 by Makarov and Vinberg involved only few other Lie groups, thus adding credit to conjectures of Selberg and Pyatetski-Shapiro to the effect that “for most semisimple Lie groups” discrete subgroups of finite covolume are necessarily arithmetic. Margulis’s most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question.

Margulis’ arithmeticity theorem and related rigidity properties for irreducible lattices of semisimple Lie groups of higher rank might not be expected by Poincaré.