## 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.

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## References

Aleksandrov, P.S. 1972. Poincaré and topology. With an appendix by V. K. Zoric. Uspehi Mat. Nauk 27:no. 1(163), 147–158. It is included in (Browder 1983).

Arnold, V. 1963. Small denominators and problems of stability of motion in classical and celestial mechanics.

*Russian Mathematical Surveys*18 (6114): 91–192.Arnold, V. 2006. The underestimated Poincaré: Forgotten and neglected theories of Poincaré.

*Russian Mathematical Surveys*61 (1): 1–18.Barrow-Green, J. 1997.

*Poincaré and the three body problem. History of mathematics, 11*, xvi+272. Providence: American Mathematical Society.Beardon, A. 1983.

*The geometry of discrete groups. Graduate texts in mathematics, 91*, xii+337. New York: Springer.Bollinger, M. 1972. Geschichtliche Entwicklung des Homologiebegriffs.

*Archive for History of Exact Sciences*9 (2): 94–170.Borel, A. 1969. Introduction aux groupes arithmétiques. Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341 Hermann, Paris, 125 pp.

Bottazzini, U., and J. Gray. 2013.

*Hidden harmony-geometric fantasies. The rise of complex function theory*, xviii+848., Sources and studies in the history of mathematics and physical sciences New York: Springer.Boyer, C. 1991.

*A history of mathematics*. Revised reprint of the second edition. With a foreword by Isaac Asimov. Revised and with a preface by Uta C. Merzbach. John Wiley & Sons, Inc., New York. xx+715 pp.Browder, F. 1983.

*The mathematical heritage of Henri Poincaré.*Part 1 and Part 2. Proceedings of the symposium held at Indiana University, Bloomington, Ind., April 7–10, 1980. Proceedings of Symposia in Pure Mathematics, 39. American Mathematical Society, Providence, RI. ix+435 pp., and ix+435 pp.Brown, K. 1974. Euler characteristics of discrete groups and G-spaces.

*Inventiones Mathematicae*27: 229–264.Burde, G., and H. Zieschang. 1999. Development of the concept of a complex.

*History of topology*, 103–110, North-Holland, Amsterdam.Cartan, E. 1922. Leçons sur les invariants intégraux, Paris: A. Hermann & fils, x+210 pp.

Chandler, B., and W. Magnus. 1982.

*The history of combinatorial group theory. A case study in the history of ideas*, viii+234., Studies in the history of mathematics and physical sciences, 9 New York: Springer.Charpentier, É., É. Ghys, and A. Lesne. 2010.

*The scientific legacy of Poincaré*, xiv+392., Translated from the 2006 French original by Joshua Bowman history of mathematics, 36 Providence: American Mathematical Society.Chenciner, A. 2015. Poincaré and the three-body problem.

*Henri Poincaré, 1912–2012*, pp. 51–149, Prog. Math. Phys., 67, Birkhäuser/Springer, Basel.Conway, J., and D. Smith. 2003.

*On quaternions and octonions: Their geometry, arithmetic, and symmetry, xii+159*. Natick: A K Peters Ltd.Coxeter, H.S.M. 1934. Discrete groups generated by reflections.

*Annals of Mathematics*2 (353): 588–621.de Saint-Gervais, H.P. 2014–2019.

*Analysis Situs: Topologie algébrique des variétés*, an electronic book available at http://analysis-situs.math.cnrs.fr/.de Saint-Gervais, H.P. 2016.

*Uniformization of Riemann surfaces. Revisiting a hundred-year-old theorem*. Translated from the 2010 French by Robert G. Burns. Heritage of European Mathematics. European Mathematical Society (EMS), Zurich. xxx+482 pp.Dieudonné, J. 1989.

*A history of algebraic and differential topology. 1900–1960*, xxii+648. Boston: Birkhäuser.Dieudonné, J. 1985.

*History of algebraic geometry. An outline of the history and development of algebraic geometry.*Translated from the French by Judith D. Sally. Wadsworth Mathematics Series. Wadsworth International Group, Belmont, CA. xii+186 pp.Dixon, J.D. 1971.

*Structure of linear groups*. London: Van Nostrand Reinhold.Donaldson, S. 1999. One hundred years of manifold topology.

*History of topology*, 435–447, North-Holland, Amsterdam.Duplantier, B., and V. Rivasseau. 2015.

*Henri Poincaré, 1912-2012.*Proceedings of the 16th Poincaré Seminar held in Paris, November 24, 2012. Progress in Mathematical Physics, 67. Birkhäuser/Springer, Basel. xiv+233 pp.Dyck, W. 1888. Beiträge zur Analysis situs.

*Mathematische Annalen*32 (4): 457–512.Dyck, W. 1890. Beiträge zur Analysis situs.

*Mathematische Annalen*37 (2): 273–316.Dyck, W. 1891. Ueber die gestaltichen Verhältnisse der durch eine Differentialgleichung erster Ordnung zwischen zwei Variabeln definirten Curvensysteme, Stitzungsberichte der Mathematisch-physikalische Classe der Academie der Wisenschaften zu München, Band XXI. Jahrgang.

Dyck, W. 1909.

*Über die singulären Srellen eines Systems von Differentialgleichungen ester Ordnung, Stitzungsberichte der Königlich Bayerischen Academie der Wisenschaften Mathematisch-physikalische Klasse, Jahrgang, 15, 6*. Vorgetragen am: Abhandlung.Dyck, W. 1884. On the “Analysis Situs” of 3-dimensional spaces, Report of the British Association for the Advancement of Science, p. 648.

Gilain, C. 1991. La théorie qualitative de Poincaré et le problm̀e de l’intégration des équations différentielles, in

*La France mathématique*, H. Gispert (édité par), La Société mathématique de France (1870-1914), SFHST et SMF, p. 215-242.Gordon, C., and 1960.

*3-dimensional topology up to, 1960. History of topology*, 449–489. Amsterdam: North-Holland.Gray, J. 2013.

*Henri Poincaré. A scientific biography*, xvi+592. Princeton: Princeton University Press.Gray, J. 2008.

*Linear differential equations and group theory from Riemann to Poincaré. Reprint of the 2000 second edition Modern Birkhäuser Classics*, 2nd ed, xx+338. Boston: Birkhäuser.Grattan-Guinness, I. 1998.

*The Norton history of the mathematical sciences. The rainbow of mathematics.*Norton History of Science. W. W. Norton & Co. Inc., New York. vi+817 pp.Griess, R., and A. Ryba. 1999. Finite simple groups which projectively embed in an exceptional Lie group are classified!.

*Bulletin of the American Mathematical Society (N.S.)*36 (1): 75–93.Griffiths, P. 1982. Poincaré and algebraic geometry.

*Bulletin of the American Mathematical Society (N.S.)*6 (2): 147–159.Hadamard, J. 1933. The later scientific work of Henri Poincaré. Rice University studies ; v. 20, no. 1. The Rice institute pamphlet. [Houston, Tex., 1933] 24 cm. vol. XX, no. 1, January, p. 1–86. illus.

Hadamard, J. 1922. The early scientific work of Henri Poincaré. Rice University studies ; v. 9, no. 3. Rice institute pamphlets. Houston. 23 cm. IX(3):111–183.

Herreman, A. 1997. Le statut de la géométrie dans quelques textes sur l’homologie, de Poincaré aux années 1930.

*Revue d’Histoire des Mathématiques*3 (2): 241–293.Humphreys, J. 1990.

*Reflection groups and coxeter groups. Cambridge studies in advanced mathematics, 29*, xii+204. Cambridge: Cambridge University Press.James, I. 1998. Reflections of the history of topology.

*Rendiconti del Seminario Matematico e Fisico di Milano*66 (1996): 87–96.Jordan, C. 1878. Memoire sur les equations differentielles lineaires a integrate algebrique.

*Journal für die reine und angewandte Mathematik*84: 89–215.Jordan, C. 1880. Sur la determination des groupes d’order fini contenus dans le groupe lineaire.

*Atti Della R Accademia Delle Scienze Fisiche E Matematiche VIII*11: 1–41.Jordan, C. 1868. Mémoire sur les groupes de mouvements.

*Annali di Matematica*2: 167–215.Jordan, C. 1869. Mémoire sur les groupes de mouvements.

*Annali di Mathematica*2: 322–345.Katz, V. 1993.

*A history of mathematics. An introduction*, xiv+786. New York: HarperCollins College Publishers.Katz, V. 1981. The history of differential forms from Clairaut to Poincaré.

*Historia Mathematica*8 (2): 161–188.Katz, V. 1999. Differential forms.

*History of topology*, 111–122, North-Holland, Amsterdam.Klein, F. 1875. Ueber bindre Formen mit linearen Transformationen in sich selbst.

*Mathematische Annalen*9: 183–208.Kline, M. 1990.

*Mathematical thought from ancient to modern times.*Vol. 3. Second edition. The Clarendon Press, Oxford University Press, New York. pp. i–xvi, 813–1212 and i–xxii.Margulis, G.A. 1991.

*Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17*, x+388. Berlin: Springer.Mawhin, J. 2005. Poincaré Henri. A life A life in the service of science.

*Notices of the American Mathematical Society*52 (9): 1036–1044.Mawhin, J. 2000. Poincaré early use of Analysis situs in nonlinear differential equation: Variations around the theme of Kronecker’s integral.

*Philosophia Scientiae*4 (1): 103–143.Milnor, J. 2003. Towards the Poincaré conjecture and the classification of 3-manifolds.

*Notices Amer. Math. Soc*. 50 (10): 1226–1233.Milnor, J. 2006.

*The Poincaré conjecture. The millennium prize problems*, 71–83. Cambridge: Clay Mathematics Institute.Milnor, J. 1976. Hilbert’s problem 18: On crystallographic groups, fundamental domains, and on sphere packing.

*Mathematical developments arising from Hilbert problems*(Proceedings of Symposia in Pure Mathematics, Northern Illinois Univ., De Kalb, Ill., 1974), pp. 491–506. Proceedings of Symposia in Pure Mathematics, Vol. XXVIII, American Mathematical Society, Providence, R. I.Nabonnand, P. 1999. La correspondance entre Henri Poincaré et Gösta Mittag-Leffler. Avec en annexes les lettres échangées par Poincaré avec Fredholm, Gyldén et Phragmén. Publications des Archives Henri-Poincaré. Birkhäuser Verlag, Basel. 421 pp.

Novikov, S. 2004. Henri Poincaré and XXth century topology. In

*Solvay Workshops and Symposia*, vol. 2, Symposium Henri Poincaré, edited by P. Gaspard, M. Henneaux, and F. Lambert, pp. 17–24.Picard, E., and G. Simart. 1971. Théorie des fonctions algébriques de deux variables indépendantes. Tome I, II. Réimpression corrigée (en un volume) de l’édition en deux volumes de 1897 et 1906. Chelsea Publishing Co., Bronx, N.Y. Tome I: xiv+244 pp.; Tome II: ii+52 pp.

Poincaré, H. 1892. Sur l’Aanalysis situs.

*Comptes rendus de l’Academie des Sciences*115: 633–636.Poincaré, H. 1893. Sur la généralisation d’un théorème d’Euler relatif aux polyèdres.

*C.R*117: 144–145.Poincaré, H. 1895. Analysis situs.

*Journal de l’École Polytechnique, Series*21: 1–123.Poincaré, H. 1899. Sur les nombres de Betti.

*C.R*128: 629–630.Poincaré, H. 1899. Complément à l’Analysis Situs.

*Rendiconti del Circolo Matematico di Palermo, t*. 13: 285–343.Poincaré, H. 1900. Second Complément à l’Analysis Situs.

*Proceedings of the London Mathematical Society*32: 277–308.Poincaré, H. 1901. Sur “Analysis Situs”.

*C.R*133: 707–709.Poincaré, H. 1901. Sur la connexion des surfaces algébriques.

*C.R*133: 969–973.Poincaré, H. 1902. Sur certaines surfaces algébriques. Troisième complément à l’Analysis situs.

*Bulletin de la Société de France*30: 49–70.Poincaré, H. 1902. Sur les cycles des surfaces algébriques: Quatrième complément à l’Analysis Situs.

*Journal de Mathématiques Pures et Appliquées*8 (5): 169–214.Poincaré, H. 1904. Cinquiéme complément à l’analysis situs.

*Rendiconti del Circolo Matematico di Palermo*18: 45–110.Poincaré, H. 1905. Sur les lignes géodésiques des surfaces convexes.

*Transactions of the American Mathematical Society*6: 237–274.Poincaré, H. 1910. Sur les courbes tracées sur les surfaces algébriques.

*Annales Scientifiques de l’École Normale Supérieure*27 (3): 55–108.Poincaré, H. 1912. Sur un théorème de géométrie.

*Rendicotti Matematico di Palermo*33 (33): 375–407.Poincaré, H. 1913. Mathematics and science: Last essays. Translated from the French by John W. Bolduc Dover Publications, Inc., New York 1963 iv+121 pp. (A translation of the first edition of Dernières pensées, published by Ernest Flammarion, Paris).

Poincaré, H. 2010.

*Papers on topology. Analysis situs and its five supplements.*Translated and with an introduction by John Stillwell. History of Mathematics, 37. American Mathematical Society, Providence, RI; London Mathematical Society, London. xx+228 pp.Poincaré, H. 1921. Analyse des travaux scientifiques de Poincaré faite par lui-m\(\hat{e}\)me.

*Acta Mathematica*38: 1–135.Poincaré, H. 1993.

*New methods of celestial mechanics. Vol. 3. Integral invariants and asymptotic properties of certain solutions.*Translated from the French. Revised reprint of the 1967 English translation. With endnotes by G. A. Merman. Edited and with an introduction by Daniel L. Goroff. History of Modern Physics and Astronomy, 13. American Institute of Physics, New York. pp. i–xx, 723–1078, E19–E23 and xxi–xxv.Raghunathan, M.S. 1972.

*Discrete subgroups of Lie groups.*Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. Springer, New York. ix+227 pp.Sarkaria, K.S. 1999. The topological work of Henri Poincaré.

*History of topology*, 123–167, North-Holland, Amsterdam.Sarkaria, K.S. 1994. Poincaré’s paper on topology, unpublished Notes of Lectures given in 1993–1994. http://www.kssarkaria.org/docs/Sarkaria-Analysis-Situs.pdf.

Scholz, E. 1999. The concept of manifold, 1850–1950.

*History of topology*, 25–64, North-Holland, Amsterdam.Serre, J.P. 2000. Sous-groupes finis des groupes de Lie. Séminaire Bourbaki, Vol. 1998/99. Astérisque No. 266, Exp. No. 864(5):415–430.

Siersma, D. 2012. Poincaré and Analysis situs, the beginning of algebraic topology.

*Nieuw Archief Voor Wiskunde. Serie (5)*13 (3): 196–200.Stillwell, J. 1985.

*Papers on Fuchsian functions.*Translated from the French and with an introduction by John Stillwell. Springer, New York. ii+483 pp.Stillwell, J. 2012. Poincaré and the early history of 3-manifolds.

*Bulletin of the American Mathematical Society (N.S.)*49 (4): 555–576.Tits, J. 1980. The Work of Gregori Aleksandrovitch Margulis.

*Proceedings of the International Congress of Mathematicians*(Helsinki, 1978), pp. 58-63, Acad. Sci. Fennica, Helsinki.van der Waerden, B.L. 1985.

*A history of algebra. From al-Khwarizmi to Emmy Noether*, xi+271. Berlin: Springer.Vanden Eynde, R. 1999. Development of the concept of homotopy.

*History of topology*, 65–102, North-Holland, Amsterdam.Verhulst, F. 2012.

*Henri Poincaré. Impatient genius.*Springer, New York. xii+260 pp.Verhulst, F. 2016. Henri Poincaré’s inventions in dynamical systems and topology.

*The foundations of chaos revisited: from Poincaré to recent advancements*, 1–25. Underst. Complex Syst.: Springer.Verhulst, F., and S. Wepster. 2012. Henri Poincaré Centennial Issue in

*Nieuw Archief voor Wiskunde*, September.Vinberg, E. 1981. The nonexistence of crystallographic reflection groups in Lobachevskii spaces of large dimension.

*Funktsional. Anal. i Prilozhen*. 15 (2): 67–68.Vinberg, E., and O.V. Shvartsman. 1993. Discrete groups of motions of spaces of constant curvature. Geometry, II, 139–248, Encyclopaedia Math. Sci., 29, Springer, Berlin.

Weibel, C. 1999. History of homological algebra.

*History of topology*, 797–836, North-Holland, Amsterdam.Weyl, H. 1913. Die Idee der Riemannschen Fläche. Reprint of the German original. With essays by Reinhold Remmert, Michael Schneider, Stefan Hildebrandt, Klaus Hulek and Samuel Patterson. Edited and with a preface and a biography of Weyl by Remmert. Teubner-Archiv zur Mathematik. Supplement, 5. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997. xxii+240 pp.

Zariski, O. 1995.

*Algebraic surfaces.*With appendices by S. S. Abhyankar, J. Lipman and D. Mumford. Preface to the appendices by Mumford. Reprint of the second (1971) edition. Classics in Mathematics. Springer, Berlin. xii+270 pp.

## Acknowledgements

We would like to thank Joseph Dauben, Catherine Goldstein, Jean Mawhin, Philippe Nabonnand, David Rowe, Norbert Schappacher, Kenji Ueno, Xiaofei Wang, Baoqiang Yang for their valuable comments and suggestions, and for some helpful references during writing this paper and giving a talk based on it. Particularly we would like to thank Jeremy Gray for reading carefully an earlier version of this paper and for many valuable and constructive suggestions on how to revise the paper. The first author would also like to thank Jeremy Gray for many informative and educational e-mail correspondences about several general questions on the history of mathematics during the revision of this paper.

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## Appendix on finite and discrete subgroups of Lie groups

### 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

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*(2*n*) 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.

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Ji, L., Wang, C. Poincaré’s stated motivations for topology.
*Arch. Hist. Exact Sci.* **74, **381–400 (2020). https://doi.org/10.1007/s00407-020-00247-y

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DOI: https://doi.org/10.1007/s00407-020-00247-y