Grassmann manifolds
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Contents |
1 Introduction
Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. They play a key role in topology and geometry as the universal spaces of vector bundles. See also Grassmannian
2 Construction and examples
2.1 Construction
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2.2 The canonical bundle
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The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle:
Low dimensional Grassmannians
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Embeddings of Grassmannians into affine and projective space
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Infinite dimensional Grassmannians
Infinite dimensional Grassmannians. Natural inclusions of vector space defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted and also . One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions and its colimit is denoted
Invariants
Homotopy groups
Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups do not depend on , if Homotopy groups in the stable range are described by the Bott periodicity theorem.
Proposition 11.1 {(R.Bott).} For each there are isomorphisms: and
2.3 Cohomology groups
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3 Classification/Characterization
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4 Further discussion
Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].
5 References
- [Kirillov2004] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004. MR2069175 (2005c:22001) Zbl 02121486
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
6 External links
- The Wikipedia page on Grassmannian