Topological rigidity of quasitoric manifolds


  • Vassilis Metaftsis
  • Stratos Prassidis



Quasitoric manifolds are manifolds that admit an action of the torus that is locally the same as the standard action of $T^n$ on $\mathbb{C}^n$. It is known that the quotients of such actions are nice manifolds with corners. We prove that a class of locally standard manifolds, that contains the quasitoric manifolds, is equivariantly rigid, i.e., that any manifold that is $T^n$-homotopy equivalent to a quasitoric manifold is $T^n$-homeomorphic to it.


Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, vol. 46, Academic Press, New York-London, 1972.

Bukhshtaber, V. M. and Panov, T. E., Actions of tori, combinatorial topology and homological algebra, Uspekhi Mat. Nauk 55 (2000), no. 5 (335), 3–106, preprint arXiv:math/0010073 [math.AT].

Davis, M. W., The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008.

Davis, M. W. and Januszkiewicz, T., Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451.

Freedman, M. H., The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357–453.

Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993.

Kirby, R. C. and Taylor, L. R., A survey of $4$-manifolds through the eyes of surgery, in “Surveys on surgery theory, Vol. 2”, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 387--421.

Kuroki, S., An Orlik-Raymond type classification of simply connected $6$-dimensional torus manifolds with vanishing odd-degree cohomology, Pacific J. Math. 280 (2016), no. 1, 89–114.

Masuda, M., Equivariant cohomology distinguishes toric manifolds, Adv. Math. 218 (2008), no. 6, 2005–2012.

Masuda, M. and Panov, T., On the cohomology of torus manifolds, Osaka J. Math. 43 (2006), no. 3, 711–746.

Masuda, M. and Suh, D. Y., Classification problems of toric manifolds via topology, in “Toric topology”, Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, 2008, pp. 273--286.

Moussong, G. and Prassidis, S., Equivariant rigidity theorems, New York J. Math. 10 (2004), 151–167.

Oda, T., Convex bodies and algebraic geometry: An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 15, Springer-Verlag, Berlin, 1988.

Perelman, G., The entropy formula for the Ricci flow and its geometric applications, preprint arXiv:math/0211159 [math.DG], 2002.

Perelman, G., Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint arXiv:math/0307245 [math.DG], 2003.

Perelman, G., Ricci flow with surgery on three-manifolds, preprint arXiv:math/0303109 [math.DG], 2003.

Prassidis, S. and Spieler, B., Rigidity of Coxeter groups, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2619–2642.

Rosas, E., Rigidity theorems for right angled reflection groups, Trans. Amer. Math. Soc. 308 (1988), no. 2, 837–848.

Schultz, R., On the topological classification of linear representations, Topology 16 (1977), no. 3, 263–269.

Wall, C. T. C., Surgery on compact manifolds, London Mathematical Society Monographs, no. 1, Academic Press, London-New York, 1970.

Wiemeler, M., Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism, Arch. Math. (Basel) 98 (2012), no. 1, 71–85.

Wiemeler, M., Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds, Math. Z. 273 (2013), no. 3-4, 1063–1084.

Yoshida, T., Locally standard torus fibrations, in “Proceedings of 34th Symposium on Transformation Groups”, Wing Co., Wakayama, 2007, pp. 107--118.

Yoshida, T., Local torus actions modeled on the standard representation, Adv. Math. 227 (2011), no. 5, 1914–1955.




How to Cite

Metaftsis, V., & Prassidis, S. (2018). Topological rigidity of quasitoric manifolds. MATHEMATICA SCANDINAVICA, 122(2), 179–196.