Algebraic topology of \(G_{2}\) manifolds. (English) Zbl 1334.53048
Summary: In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group \(G_2\). Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of \(G_2\) manifolds with special properties, which appear in the first named author’s work with S. Salur about \(G_2\)dualities.
References:
| [1] | Akbulut, S.; Efe, B.; Salur, S., Mirror duality in a Joyce manifold, Adv. Math., 223, 2, 444-453 (2010) · Zbl 1185.53054 |
| [2] | Akbulut, S.; Salur, S., Calibrated manifolds and gauge theory, J. Reine Angew. Math., 2008, 625, 187-214 (2008) · Zbl 1157.53027 |
| [3] | Akbulut, S.; Salur, S., Deformations in \(G_2\) Manifolds., Adv. Math., 217, 5, 2130-2140 (2008) · Zbl 1143.53047 |
| [4] | Akbulut, S.; Salur, S., Mirror Duality via \(G_2\) and Spin(7) Manifolds. Arithmetic and geometry around quantization, Progr. Math., 279, 1-21 (2010) · Zbl 1211.53098 |
| [6] | Borel, A., Topics in the homology theory of fiber bundles, (Lectures given at the University of Chicago, 1954. Notes by E. Halpern (1967), Springer-Verlag) |
| [7] | Borel, A.; Hirzebruch, F., Characteristic classes and homogeneous spaces. I, Amer. J. Math., 80, 458-538 (1958) · Zbl 0097.36401 |
| [9] | Dummit, D.; Foote, R., Abstract Algebra (2004), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. Hoboken, NJ, xii+932 pp. ISBN: 0-471-43334-9 · Zbl 1037.00003 |
| [10] | Gluck, H.; Mackenzie, D.; Morgan, F., Volume-minimizing cycles in Grassmann manifolds, Duke Math. J., 79, 2, 335-404 (1995) · Zbl 0837.53035 |
| [11] | Greub, W.; Halperin, S.; Vanstone, R., Connections, curvature, and cohomology. Volume III: Cohomology of principal bundles and homogeneous spaces, (Pure and Applied Mathematics, vol. 47-III (1976), Academic Press: Academic Press New York-London), xxi+593 pp · Zbl 0372.57001 |
| [12] | Guillemin, V.; Pollack, A., Differential Topology (2010), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI, xviii+224 pp., ISBN: 978-0-8218-5193-7. Reprint of the 1974 original · Zbl 1420.57001 |
| [13] | Harris, J., Algebraic geometry. A first course, (Graduate Texts in Mathematics, vol. 133 (1992), Springer-Verlag: Springer-Verlag New York), xx+328 pp · Zbl 0779.14001 |
| [14] | Harvey, F. R.; Lawson, H. B., Calibrated geometries, Acta. Math., 148, 47-157 (1982) · Zbl 0584.53021 |
| [15] | Hatcher, A., Algebraic Topology (2002), Cambridge University Press: Cambridge University Press Cambridge, xii+544 pp. ISBN: 0-521-79160-X · Zbl 1044.55001 |
| [18] | Milnor, J.; Stasheff, J., Characteristic classes, (Annals of Mathematics Studies, No. 76 (1974), Princeton University Press: Princeton University Press Princeton, NJ), vii+331 pp · Zbl 0298.57008 |
| [19] | Mimura, M.; Toda, H., (Topology of Lie Groups I and II. Topology of Lie Groups I and II, Translations of Mathematical Monographs (1991), American Mathematical Society) · Zbl 0757.57001 |
| [20] | Paechter, G. F., The groups \(\pi_r(V_{n, m})\), I. Quart. J. Math. Oxford Ser. (2), 7, 249-268 (1956) · Zbl 0119.18603 |
| [21] | Sato, Hajime, Algebraic topology: an intuitive approach, (Translations of Mathematical Monographs, vol. 183 (1999), AMS: AMS Providence, RI), xviii+118 pp., ISBN: 0-8218-1046-4 · Zbl 0916.55001 |
| [22] | Shi, Jin; Zhou, Jianwei, Characteristic classes on Grassmannians., Turkish J. Math., 38, 3, 492-523 (2014) · Zbl 1298.14054 |
| [23] | Spanier, E., Algebraic Topology (1981), Springer-Verlag: Springer-Verlag New York, Berlin, xvi+528 pp., ISBN: 0-387-90646-0. Corrected reprint |
| [24] | Stiefel, E., Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten, Comment. Math. Helv., 8, 1, 305-353 (1935), (in German) · Zbl 0014.41601 |
| [25] | Ünal, İ., Topology of \(\phi \)-convex domains in calibrated manifolds, Bull. Braz. Math. Soc. (N.S.), 42, 2, 259-275 (2011) · Zbl 1243.53089 |
| [26] | Whitehead, J. H.C., On the groups \(\pi_r(V_{n, m})\) and sphere-bundles. Corrigendum., Proc. London Math. Soc. (2), 49, 479-481 (1947) |
| [27] | Zhou, Jianwei, Morse functions on Grassmann manifolds., Proc. Roy. Soc. Edinburgh Sect. A, 135, 1, 209-221 (2005) · Zbl 1078.53047 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
