Publication: On SO(3)-bundles over the Wolf spaces
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2019
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Abstract
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.
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[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, https://nms.kcl.ac.uk/simon.salamon/T/luminy.pdf
(see also the conference talks https://nms.kcl.ac.uk/simon.salamon/T/ober.pdf and
http://calvino.polito.it/esalamon/T/levico.pdf)
[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.