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An obstruction to the existence of real projective structures
Topology and its Applications 265 (2019) 106828
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
An obstruction to the existence of real projective structures Hatice Çoban Middle East Technical University, Department of Mathematics, Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 23 July 2019 Received in revised form 24 July 2019 Accepted 24 July 2019 Available online 29 July 2019
a b s t r a c t In this short note, we give an obstruction to obtain examples of higher dimensional manifolds with infinite fundamental groups, including the infinite cyclic group Z, admitting no real projective structure. © 2019 Elsevier B.V. All rights reserved.
MSC: 57N16 57S25 53A20 Keywords: Real projective structure Developing map and holonomy
1. Introduction An Ehresmann structure is an (X, G) structure where X is a locally homogeneous space and the Lie group G acts transitively on X (see [4], [5], [6], [8] for more information about (X, G) structures). The pair (dev, hol) is called a developing pair for the geometric structure (X, G) ([9]). A real projective structure on a manifold M n is an (RP n , P GL(n + 1, R)) structure. It is well known that every surface has a real projective structure and the classification of these structures are studied ([2]). D. Cooper and W. Goldman gave an example of a 3-dimensional manifold with the fundamental group Z2 ∗ Z2 not admitting a real projective structure and this is the first known example ([3]). If a simply connected closed manifold admits a real projective structure then it must be a sphere. Therefore, any simply connected manifold M which is not a sphere does not admit a real projective structure. Since there are examples of simply connected manifolds that are not spheres in dimension bigger than 3 (e.g. complex projective n-space) there are many higher dimensional manifolds that do not admit a real projective structure. On the other hand, it is not easy to show a manifold with infinite fundamental group does not have a real projective structure. In this note, we give examples of higher dimensional manifolds E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2019.106828 0166-8641/© 2019 Elsevier B.V. All rights reserved.
H. Çoban / Topology and its Applications 265 (2019) 106828
2
with infinite fundamental groups, including the infinite cyclic group Z, which do not admit a real projective structure. Acknowledgements. I am grateful to my advisor Yıldıray Ozan for his support throughout this work and the anonymous referee for his or her comments and suggestions. 2. Main result In this section, we will give an obstruction to obtain examples of manifolds with infinite fundamental groups, including the infinite cyclic group Z, admitting no real projective structure. Theorem 2.1. Let M n be a smooth manifold containing a smooth simply connected submanifold Q, which does not admit any immersion into Rn . Then M does not have any real projective structure. Proof. Assume that M admits a real projective structure. Then there exists a developing map such that M
dev
RP n ,
M denotes the universal cover of M . Note that since Q is simply connected Q is contained, as a where M . submanifold, in the universal cover M Consider the following diagram Sn
M
dev
RP n
where the horizontal map lifts to S n , because Q is simply connected. −→ RP n is an immersion and the double cover S n −→ RP n is a local Moreover, since the map M diffeomorphism, the lift of the restriction of the developing map to the submanifold Q, Q −→ S n is also an immersion. Moreover, Q −→ S n \ {p} = Rn is an immersion where p is a point in S n , which is not in the image of Q. However, this is a contradiction to the assumption. 2 Now we can state a corollary of the above result: Corollary 2.2. Let M n be a simply connected manifold which does not admit any immersion into Rn+1 . Then M × S 1 does not have any real projective structure.
H. Çoban / Topology and its Applications 265 (2019) 106828
3
Before we pass to consequences of this observation we recall a well known result, which follows from the general properties of Pontryagin classes. Theorem 2.3. ([7], Chapter 4) If there is an immersion M n −→ Rn+1 , where M is an orientable manifold then the Pontryagin classes pi (M n ) are all two torsion, for i ≥ 1. Example. Let M = CP 2 . The first Pontryagin class of CP 2 is p1 = c21 −2c2 , where ci ’s are the Chern classes, for i = 1, 2. Then p1 = c21 − 2c2 = 9 − 2.3 = 3. Hence p1 is not a torsion class. By Theorem 2.3, there is no immersion CP 2 −→ R5 . Therefore, CP 2 × S 1 does not have a real projective structure. Remark 1. Note that since S n−1 has an immersion into Rn , the above theorem does not imply that RP n #RP n can not have a real projective structure. However, in 3-dimension, as mentioned in the introduction the manifold RP 3 #RP 3 admits no real projective structure. The author generalized this result to higher dimensions in her thesis ([1]). References [1] Hatice Çoban, Smooth Manifolds With Infinite Fundamental Group Admitting No Real Projective Structure, Ph.D Thesis, 2017. [2] Suhyoung Choi, William M. Goldman, The classification of real projective structures on compact surfaces, Bull. Am. Math. Soc. (N.S.) 34 (2) (1997) 161–171. [3] Daryl Cooper, William Goldman, A 3-manifold with no real projective structure, Ann. Fac. Sci. Toulouse Math. (6) 24 (5) (2015) 1219–1238. [4] William M. Goldman, Geometric structures on manifolds and varieties of representations, in: Geometry of Group Representations, Boulder, CO, 1987, in: Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 169–198. [5] William M. Goldman, What is. . .a projective structure?, Not. Am. Math. Soc. 54 (1) (2007) 30–33. [6] William M. Goldman, Locally homogeneous geometric manifolds, in: Proceedings of the International Congress of Mathematicians, vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 717–744. [7] John W. Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, Princeton University Press, 1974, pp. 5–89, 155–156, Publ. Inst. Math. Univ. Strasbourg 11. [8] John G. Ratcliffe, Foundations of Hyperbolic Manifolds, second edition, Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. [9] William P. Thurston, Three-Dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997, edited by Silvio Levy.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
An obstruction to the existence of real projective structures Hatice Çoban Middle East Technical University, Department of Mathematics, Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 23 July 2019 Received in revised form 24 July 2019 Accepted 24 July 2019 Available online 29 July 2019
a b s t r a c t In this short note, we give an obstruction to obtain examples of higher dimensional manifolds with infinite fundamental groups, including the infinite cyclic group Z, admitting no real projective structure. © 2019 Elsevier B.V. All rights reserved.
MSC: 57N16 57S25 53A20 Keywords: Real projective structure Developing map and holonomy
1. Introduction An Ehresmann structure is an (X, G) structure where X is a locally homogeneous space and the Lie group G acts transitively on X (see [4], [5], [6], [8] for more information about (X, G) structures). The pair (dev, hol) is called a developing pair for the geometric structure (X, G) ([9]). A real projective structure on a manifold M n is an (RP n , P GL(n + 1, R)) structure. It is well known that every surface has a real projective structure and the classification of these structures are studied ([2]). D. Cooper and W. Goldman gave an example of a 3-dimensional manifold with the fundamental group Z2 ∗ Z2 not admitting a real projective structure and this is the first known example ([3]). If a simply connected closed manifold admits a real projective structure then it must be a sphere. Therefore, any simply connected manifold M which is not a sphere does not admit a real projective structure. Since there are examples of simply connected manifolds that are not spheres in dimension bigger than 3 (e.g. complex projective n-space) there are many higher dimensional manifolds that do not admit a real projective structure. On the other hand, it is not easy to show a manifold with infinite fundamental group does not have a real projective structure. In this note, we give examples of higher dimensional manifolds E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2019.106828 0166-8641/© 2019 Elsevier B.V. All rights reserved.
H. Çoban / Topology and its Applications 265 (2019) 106828
2
with infinite fundamental groups, including the infinite cyclic group Z, which do not admit a real projective structure. Acknowledgements. I am grateful to my advisor Yıldıray Ozan for his support throughout this work and the anonymous referee for his or her comments and suggestions. 2. Main result In this section, we will give an obstruction to obtain examples of manifolds with infinite fundamental groups, including the infinite cyclic group Z, admitting no real projective structure. Theorem 2.1. Let M n be a smooth manifold containing a smooth simply connected submanifold Q, which does not admit any immersion into Rn . Then M does not have any real projective structure. Proof. Assume that M admits a real projective structure. Then there exists a developing map such that M
dev
RP n ,
M denotes the universal cover of M . Note that since Q is simply connected Q is contained, as a where M . submanifold, in the universal cover M Consider the following diagram Sn
M
dev
RP n
where the horizontal map lifts to S n , because Q is simply connected. −→ RP n is an immersion and the double cover S n −→ RP n is a local Moreover, since the map M diffeomorphism, the lift of the restriction of the developing map to the submanifold Q, Q −→ S n is also an immersion. Moreover, Q −→ S n \ {p} = Rn is an immersion where p is a point in S n , which is not in the image of Q. However, this is a contradiction to the assumption. 2 Now we can state a corollary of the above result: Corollary 2.2. Let M n be a simply connected manifold which does not admit any immersion into Rn+1 . Then M × S 1 does not have any real projective structure.
H. Çoban / Topology and its Applications 265 (2019) 106828
3
Before we pass to consequences of this observation we recall a well known result, which follows from the general properties of Pontryagin classes. Theorem 2.3. ([7], Chapter 4) If there is an immersion M n −→ Rn+1 , where M is an orientable manifold then the Pontryagin classes pi (M n ) are all two torsion, for i ≥ 1. Example. Let M = CP 2 . The first Pontryagin class of CP 2 is p1 = c21 −2c2 , where ci ’s are the Chern classes, for i = 1, 2. Then p1 = c21 − 2c2 = 9 − 2.3 = 3. Hence p1 is not a torsion class. By Theorem 2.3, there is no immersion CP 2 −→ R5 . Therefore, CP 2 × S 1 does not have a real projective structure. Remark 1. Note that since S n−1 has an immersion into Rn , the above theorem does not imply that RP n #RP n can not have a real projective structure. However, in 3-dimension, as mentioned in the introduction the manifold RP 3 #RP 3 admits no real projective structure. The author generalized this result to higher dimensions in her thesis ([1]). References [1] Hatice Çoban, Smooth Manifolds With Infinite Fundamental Group Admitting No Real Projective Structure, Ph.D Thesis, 2017. [2] Suhyoung Choi, William M. Goldman, The classification of real projective structures on compact surfaces, Bull. Am. Math. Soc. (N.S.) 34 (2) (1997) 161–171. [3] Daryl Cooper, William Goldman, A 3-manifold with no real projective structure, Ann. Fac. Sci. Toulouse Math. (6) 24 (5) (2015) 1219–1238. [4] William M. Goldman, Geometric structures on manifolds and varieties of representations, in: Geometry of Group Representations, Boulder, CO, 1987, in: Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 169–198. [5] William M. Goldman, What is. . .a projective structure?, Not. Am. Math. Soc. 54 (1) (2007) 30–33. [6] William M. Goldman, Locally homogeneous geometric manifolds, in: Proceedings of the International Congress of Mathematicians, vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 717–744. [7] John W. Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, Princeton University Press, 1974, pp. 5–89, 155–156, Publ. Inst. Math. Univ. Strasbourg 11. [8] John G. Ratcliffe, Foundations of Hyperbolic Manifolds, second edition, Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. [9] William P. Thurston, Three-Dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997, edited by Silvio Levy.