A 3–Manifold with no Real Projective Structure
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, p. 1219-1238

We show that the connected sum of two copies of real projective 3-space does not admit a real projective structure. This is the first known example of a connected 3-manifold without a real projective structure.

Nous montrons que la somme connexe de deux copies de l’espace projectif de dimension trois n’admet pas de structure projective réelle. Ceci est le premier exemple connu d’une variété connexe de dimension 3 sans structure projective réelle.

@article{AFST_2015_6_24_5_1219_0,
     author = {Cooper, Daryl and Goldman, William},
     title = {A 3--Manifold with no Real Projective Structure},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {5},
     year = {2015},
     pages = {1219-1238},
     doi = {10.5802/afst.1482},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2015_6_24_5_1219_0}
}
Cooper, Daryl; Goldman, William. A 3–Manifold with no Real Projective Structure. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, pp. 1219-1238. doi : 10.5802/afst.1482. http://www.numdam.org/item/AFST_2015_6_24_5_1219_0/

[1] Benoist (Y.).— Nilvariétés projectives. Comment. Math. Helv., 69(3), p. 447-473 (1994). | MR 1289337 | Zbl 0839.53033

[2] Benoist (Y.).— Tores affines. In Crystallographic groups and their generalizations (Kortrijk, 1999), volume 262 of Contemp. Math., pages 1-37. Amer. Math. Soc., Providence, RI (2000). | MR 1796124 | Zbl 0990.53053

[3] Benoist (Y.).— Convexes divisibles. IV. Structure du bord en dimension 3. Invent. Math., 164(2), p. 249-278 (2006). | MR 2218481 | Zbl 1107.22006

[4] Bergeron (N.) and Gelander (T.).— A note on local rigidity. Geom. Dedicata, 107, p. 111-131 (2004). | MR 2110758 | Zbl 1062.22028

[5] Bonahon (F.).— Geometric structures on 3-manifolds. In Handbook of geometric topology, pages 93-164. North-Holland, Amsterdam (2002). | MR 1886669 | Zbl 0997.57032

[6] Canary (R. D.), Epstein (D. B. A.), and Green (P.).— Notes on notes of Thurston. In Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), volume 111 of London Math. Soc. Lecture Note Ser., p. 3-92. Cambridge Univ. Press, Cambridge (1987). | MR 903849 | Zbl 0612.57009

[7] Carrière (Y.), Dal’bo (F.), and Meigniez (G.).— Inexistence de structures affines sur les fibrés de Seifert. Math. Ann., 296(4), p. 743-753 (1993). | MR 1233496 | Zbl 0793.57006

[8] Choi (S.).— Convex decompositions of real projective surfaces. III. For closed or nonorientable surfaces. J. Korean Math. Soc., 33(4), p. 1139-1171 (1996). | MR 1424213 | Zbl 0958.53022

[9] Choi (S.) and Goldman (W. M.).— The classification of real projective structures on compact surfaces. Bull. Amer. Math. Soc. (N.S.), 34(2), p. 161-171 (1997). | MR 1414974 | Zbl 0866.57001

[10] Choi (S.) and Goldman (W. M.).— The deformation spaces of convex RP 2 -structures on 2-orbifolds. Amer. J. Math., 127(5), p. 1019-1102 (2005). | MR 2170138 | Zbl 1086.57015

[11] Cooper (D.), Long (D.), and Thistlethwaite (M.).— Computing varieties of representations of hyperbolic 3-manifolds into SL(4, R). Experiment. Math., 15(3), p. 291-305 (2006). | MR 2264468 | Zbl 1117.57016

[12] Cooper (D.), D. Long (D.), and Thistlethwaite (M. B.).— Flexing closed hyperbolic manifolds. Geom. Topol., 11, p. 2413-2440 (2007). | MR 2372851 | Zbl 1142.57009

[13] Earle (C. J.).— On variation of projective structures. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 87-99. Princeton Univ. Press, Princeton, N.J. (1981). | MR 624807 | Zbl 0474.30036

[14] Ehresmann (C.).— Variétes localement projectives. L’Enseignement Mathématique, 35, p. 317-333 (1937). | Zbl 0015.39404

[15] Goldman (W. M.).— Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3- manifolds. Trans. Amer. Math. Soc., 278(2), p. 573-583 (1983). | MR 701512 | Zbl 0518.53041

[16] Goldman (W. M.).— Geometric structures on manifolds and varieties of representations. In Geometry of group representations (Boulder, CO, 1987), volume 74 of Contemp. Math., pages 169-198. Amer. Math. Soc., Providence, RI (1988). | MR 957518 | Zbl 0659.57004

[17] Goldman (W. M.).— Convex real projective structures on compact surfaces. J. Differential Geom., 31(3), p. 791-845 (1990). | MR 1053346 | Zbl 0711.53033

[18] Goldman (W. M.).— Locally homogeneous geometric manifolds. In Proceedings of the International Congress of Mathematicians. Volume II, p. 717-744, New Delhi (2010). Hindustan Book Agency. | MR 2827816 | Zbl 1234.57001

[19] Guichard (O.) and Wienhard (A.).— Topological invariants of Anosov representations. J. Topol., 3(3), p. 578-642 (2010). | MR 2684514 | Zbl 1225.57012

[20] Hejhal (D. A.).— Monodromy groups and linearly polymorphic functions. Acta Math., 135(1), p. 1-55 (1975). | MR 463429 | Zbl 0333.34002

[21] Hubbard (J. H.).— The monodromy of projective structures. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 257-275. Princeton Univ. Press, Princeton, N.J. (1981). | MR 624819 | Zbl 0475.32008

[22] Lok (W. L.).— Deformations of locally homogeneous spaces and Kleinian groups. ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)-Columbia University. | MR 2633813

[23] Molnár (E.).— The projective interpretation of the eight 3-dimensional homogeneous geometries. Beiträge Algebra Geom., 38(2), p. 261-288 (1997). | MR 1473106 | Zbl 0889.51021

[24] Reeb (G.).— Sur certaines propriétés globales des trajectoires de la dynamique, dues à l’existence de l’invariant intégral de M. Elie Cartan. In Colloque de Topologie de Strasbourg, 1951, no. III, page 7. La Bibliothèque Nationale et Universitaire de Strasbourg (1952). | MR 52181 | Zbl 0049.18503

[25] Scott (P.).— The geometries of 3-manifolds. Bull. London Math. Soc., 15(5), p. 401-487 (1983). | MR 705527 | Zbl 0561.57001

[26] Sharpe (R. W.).— Differential geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, New York (1997). Cartan’s generalization of Klein’s Erlangen program, With a foreword by S. S. Chern. | MR 1453120 | Zbl 0876.53001

[27] Thiel (B.).— Einheitliche Beschreibung der acht Thurstonschen Geometrien. Diplomarbeit, Universitat zu Gottingen (1997).

[28] Thurston (W.).— The geometry and topology of 3-manifolds. Mimeographed notes (1979).

[29] Thurston (W. P.).— Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. | MR 1435975 | Zbl 0873.57001

[30] Waldhausen (F.).— Heegaard-Zerlegungen der 3-Sphäre. Topology, 7, p. 195-203 (1968). | MR 227992 | Zbl 0157.54501

[31] Weil (A.).— On discrete subgroups of Lie groups. Ann. of Math. (2), 72, p. 369-384 (1960). | MR 137792 | Zbl 0131.26602