On SO(3)-bundles over the Wolf spaces

Thumbnail Image
Official URL
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
We study the formality of the total space of principal SU(2) and SO(3)bundles over a Wolf space, that is a symmetric positive quaternionic K¨ahler manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose total space is non-formal.
[1] D. V. Alekseevski, Classi�cation of quaternionic spaces with solvable group of motions, Math. USSR-Izv. 9 (1975), 297-339. [2] M. Amann, Positive Quaternion K�ahler Manifolds, Ph. D. thesis, Wilhelms-Universit�at Münster, 2009. [3] M. Amann and V. Kapovitch, On �brations with formal elliptic �bers, Adv. Math. 231 (2012), 2048-2068. [4] M. Amann, Non-formal homogeneous spaces, Math. Z. 274 (2013), 1299-1325. [5] I. Biswas, M. Fernández, V. Muñoz and A. Tralle, On formality of Sasakian manifolds, J. Topol. 9 (2016), 161-180. [6] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, volume 203 of Progress in Mathematics. Birkh�auser Boston, Inc., Boston, MA, second edition, 2010. [7] A. Borel, Sur la cohomologie des espaces �fibrés principaux et des espaces homogèntes de groupes de Lie compacts, Ann. Math. 57 (1953), 115-207. [8] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958), 458-538. [9] C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Univ. Press, Oxford, 2007. [10] C. P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, 123-184, Surv. Differ. Geom., VI, Int. Press, Boston, MA, 1999. [11] C. P. Boyer, K. Galicki and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183-220. [12] C. P. Boyer, K. Galicki, B. M. Mann and E. G. Rees, Compact 3-Sasakian 7-manifolds with arbitrary second Betti number, Invent. Math. 131 (1998), 321-344. [13] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Math. vol. 98, Springer, 1985. [14] E. Cartan, Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple. (French) Ann. Sci. École Norm. Sup. 44 (1927), 345-467. [15] D. Crowley and J. Nordstr�om, The rational homotopy type of (n-1)-connected (4n-1)-manifolds, arxiv: 1505.04184v1 [math.AT]. [16] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245{274. [17] Y. Félix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Springer, 2002. [18] M. Fernández, S. Ivanov and V. Muñoz, Formality of 7-dimensional 3-Sasakian manifolds, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., arxiv: 1511.08930 [math.DG]. [19] M. Fernández and V. Muñoz, Formality of Donaldson submanifolds, Math. Z. 250 (2005), 149-175. [20] K. Galicki, S. Salamon, On Betti Numbers of 3-Sasakian Manifolds, Geom. Ded. 63 (1996), 45-68. [21] W. Greub, S. Halperin and R. Vanstone, Connections, curvature, and cohomology, Academic Press, New York, 1976. [22] P. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Progress in Math. 16, Birkh�auser, 1981. [23] S. Halperin, Lectures on minimal models, Mém. Soc. Math. France 230, 1983. [24] S. Ishihara, Quaternion K�ahler manifolds, J. Differ. Geom. 9 (1974), 483-500. [25] S. Ishihara, M. Konishi, Fibred Riemannian spaces with Sasakian 3-structure, Differential Geom., in honor of K. Yano, Kinokuniya, Tokyo 1972, 179-194. [26] K. Ishitoya, Integral cohomology ring of the symmetric space EII, J. Math. Kyoto Univ. 17 (1977), 375-397. [27] K. Ishitoya and H. Toda, On the cohomology of irreducible symmetric spaces of exceptional type, J. Math. Kyoto Univ. 17 (1977), 225-243. [28] M. Karoubi, K-theory. An introduction, Grundlehren der Mathematischen Wissenschaften 226, Springer,1978. xviii+308 pp. [29] T. Kashiwada, A note on Riemannian space with Sasakian 3-structure, Nat. Sci. Reps. Ochanomizu Univ., 22 (1971), 1-2. [30] M. Konishi, On manifolds with Sasakian 3-structure over quaternion Kaehler manifolds, Kodai Math. Sem. Rep. 26 (1975), 194-200. [31] C. LeBrun and S. Salamon, Strong rigidity of positive quaternion-K�ahler manifolds, Invent. Math. 118 (1994), 109-132. [32] J. Milnor, J. Stasheff, Characteristic classes, Annals of Mathematics Studies. Princeton University Press, 1974. [33] G. Lupton, Variations on a conjecture of Halperin, in: Homotopy and Geometry, Warsaw 1997, Banach Center Publ. 45 (1998), 115-135. [34] V. Muñoz, A. Tralle, Simply connected K-contact and Sasakian manifolds of dimension 7, Math. Z. 281 (2015), 457-470. [35] T. Nagano and M. Takeuchi, Cohomology of quaternionic K�ahler manifolds, J. Fac. Sci. Univ. Tokyo Sect IA Math. 34 (1987), 57-63. [36] M. Nakagawa, The mod 2 cohomology ring of the symmetric space EV I, J. Math. Kyoto Univ. 41 (2001), 535-556. [37] M. Nakagawa, The integral cohomology ring of E8 =T , Proc. Japan Acad. 86 (2010), 64-68. [38] J. Neisendorfer and T.J. Miller, Formal and coformal spaces, Illinois. J. Math. 22 (1978), 565-580. [39] P. Piccinni, On the cohomology of some exceptional symmetric spaces, arXiv:1609.06881 [math.DG]. [40] A. Roig and M. Saralegi, Minimal models for non-free circle actions, Illinois J. Math. 44 (2000),784-820. [41] S. Salamon, Quaternionic K�ahler manifolds, Invent. Math. 67 (1982), 143-171. [42] S. Salamon, Index theory ans special geometries, Luminy Meeting Spin Geometry and Analysis on Manifolds, Oct. 2014, (see also the conference talks and [43] D. Sullivan, Infi�nitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1978), 269-331. [44] S. Tanno, Remarks on a triple of K-contact structures, Tôhoku Math. J. 48 (1996), 519-531. [45] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, manifolds with positive �rst Chern class, J. Math. Mech. 14 (1965), 1033-1047.