On the KO–groups of toric manifolds

Cai, Li¹ ; Choi, Suyoung² ; Park, Hanchul³

¹Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou, China
²Department of Mathematics, Ajou University, Suwon, South Korea
³Department of Mathematics Education, Jeju National University, Jeju-si, South Korea

Algebraic and Geometric Topology, Tome 20 (2020) no. 5, pp. 2589-2607

Résumé

We consider the real topological K–groups of a toric manifold M, which turns out to be closely related to the topology of the small cover Mℝ, the fixed points under the canonical conjugation on M. Following the work of Bahri and Bendersky (2000), we give an explicit formula for the KO–groups of toric manifolds, and then we characterize the two extreme classes of toric manifolds according to their mod 2 cohomology groups as 𝒜(1)–modules.

Détail

DOI : 10.2140/agt.2020.20.2589

Classification : 14M25, 19E20, 55N15, 57N65
Keywords: KO–theory, toric manifolds, quasitoric manifolds

Bibliographie

[1] L Astey, A Bahri, M Bendersky, F R Cohen, D M Davis, S Gitler, M Mahowald, N Ray, R Wood, The KO∗–rings of BTm, the Davis–Januszkiewicz spaces and certain toric manifolds, J. Pure Appl. Algebra 218 (2014) 303 | DOI

[2] A Bahri, M Bendersky, The KO–theory of toric manifolds, Trans. Amer. Math. Soc. 352 (2000) 1191 | DOI

[3] A Bahri, M Bendersky, F R Cohen, S Gitler, Operations on polyhedral products and a new topological construction of infinite families of toric manifolds, Homology Homotopy Appl. 17 (2015) 137 | DOI

[4] A M Bloch, H Flaschka, T Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra, Duke Math. J. 61 (1990) 41 | DOI

[5] V M Buchstaber, T E Panov, Torus actions and their applications in topology and combinatorics, 24, Amer. Math. Soc. (2002) | DOI

[6] V M Bukhshtaber, N Raĭ, Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk 53 (1998) 139 | DOI

[7] L Cai, S Choi, Integral cohomology groups of real toric manifolds and small covers, preprint (2016)

[8] S Choi, H Park, On the cohomology and their torsion of real toric objects, Forum Math. 29 (2017) 543 | DOI

[9] Y Civan, N Ray, Homotopy decompositions and K–theory of Bott towers, –Theory 34 (2005) 1 | DOI

[10] M W Davis, Some aspherical manifolds, Duke Math. J. 55 (1987) 105 | DOI

[11] M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417 | DOI

[12] M Fujii, KO–groups of projective spaces, Osaka Math. J. 4 (1967) 141

[13] S Gitler, S López De Medrano, Intersections of quadrics, moment-angle manifolds and connected sums, Geom. Topol. 17 (2013) 1497 | DOI

[14] Y Nishimura, The quasi KO–types of certain toric manifolds, from: "Toric topology" (editors M Harada, Y Karshon, M Masuda, T Panov), Contemporary Mathematics 460, Amer. Math. Soc. (2008) 287 | DOI

[15] H Park, Wedge operations and doubling operations of real toric manifolds, Chin. Ann. Math. Ser. B 38 (2017) 1321 | DOI

[16] C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, 69, Springer (1972) | DOI

[17] C Tomei, The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J. 51 (1984) 981 | DOI

[18] A. Trevisan, Generalized Davis–Januszkiewicz spaces and their applications in algebra and topology, PhD thesis, Vrije Universiteit Amsterdam (2012)

[19] M Zibrowius, KO–rings of full flag varieties, Trans. Amer. Math. Soc. 367 (2015) 2997 | DOI