## Osaka Journal of Mathematics

### Results on the topology of generalized real Bott manifolds

#### Abstract

Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh in [4]. In this article we extend the results of [7] on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a combinatorial characterization for orientablity and spin structure.

#### Article information

Source
Osaka J. Math., Volume 56, Number 3 (2019), 441-458.

Dates
First available in Project Euclid: 16 July 2019

https://projecteuclid.org/euclid.ojm/1563242418

Mathematical Reviews number (MathSciNet)
MR3981299

Zentralblatt MATH identifier
07108025

Subjects
Primary: 55R99: None of the above, but in this section
Secondary: 57S25: Groups acting on specific manifolds

#### Citation

Dsouza, Raisa; Uma, V. Results on the topology of generalized real Bott manifolds. Osaka J. Math. 56 (2019), no. 3, 441--458. https://projecteuclid.org/euclid.ojm/1563242418

#### References

• C. Allday, M. Masuda and P. Sankaran: Transformation groups: symplectic torus actions and toric manifolds, Hindustan Book Agency, 2005.
• K. Bencsath, B. Fine, A.M. Gaglione, A.G. Myasnikov, F. Roehl, G. Rosenberger and D. Spellman: Aspects of the theory of free groups; in Algorithmic Problems in Groups and Semigroups, Springer, 51–90.
• S. Choi, M. Masuda and S.I. Oum: Classification of real Bott manifolds and acyclic digraphs, Trans. Amer. Math. Soc. 369 (2017), 2987–3011.
• S. Choi, M. Masuda and D.Y. Suh: Quasitoric manifolds over a product of simplices, Osaka Jour. Math. 47 (2010), 109–129.
• S. Choi, M. Masuda and D.Y. Suh: Topological classification of generalized Bott towers, Trans. Amer. Math. Soc. 362 (2010), 1097–1112.
• M.W. Davis and T. Januszkiewicz: Convex polytopes, coxeter orbifolds and torus actions, Duke Math. J 62 (1991), 417–451.
• R. Dsouza and V. Uma: Some results on the topology of real Bott towers, arXiv:1609.05630, (2016).
• R. Dsouza and V. Uma: A characterization of spin structure on real Bott towers, arXiv:1704.02177v1, (2017).
• A. Gąsior: Spin-structures on real Bott manifolds, J Korean Math. Soc. 54 (2017), 507–516.
• M. Grossberg and Y. Karshon: Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), 23–58.
• D. Husemoller: Fibre bundles, Springer, 1966.
• J. Jurkiewicz: Torus embeddings, polyhedra, k*-actions and homology, Dissertationes Math. (rozprawy mat.) 236 (1985), 64pp.
• Y. Kamishima and M. Masuda: Cohomological rigidity of real Bott manifolds, Algebr. Geom. Topol. 9 (2009), 2479–2502.
• S. Kuroki and Z. Lü: Projective bundles over small covers and the bundle triviality problem, Forum Math. 28 (2016), 761–781.
• S. Kuroki, M. Masuda and L. Yu: Small covers, infra-solvmanifolds and curvature, Forum Math. 27 (2015), 2981–3004.
• J.B. Lee and M. Masuda: Topology of iterated $S^1$-bundles, Osaka J. Math. 50 (2013), 847–869.
• H. Nakayama and Y. Nishimura: The orientability of small covers and coloring simple polytopes, Osaka J. Math. 42 (2005), 243–256.
• T. Oda: Convex bodies and algebraic geometry, Springer-Verlag, Berlin, 1988.
• V. Uma: On the fundamental group of real toric varieties, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), 15–31.