K-theory of oriented flag manifolds
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Vimala Ramani
Abstract
We compute the complex K-ring of oriented flag manifolds F̃(n1, … , nk), k ≥ 3, of type (n1, … , nk). We use the representation theory of spinor groups and the Hodgkin spectral sequence for the computation of the K-ring.
Acknowledgement
The author is indebted to Professor Parameswaran Sankaran for very helpful discussions and encouragement, to Shilpa Gondhali for discussions and her suggestion to simplify the notations. The author thanks the anonymous referee for very valuable comments and suggestions which improved the paper. The author thanks The Institute of Mathematical Sciences, Chennai, for the facilities provided under the Associateship Programme for College and University teachers.
(Communicated by Tibor Macko)
References
[1] Antoniano, E.—Gitler, S.—Ucci, J.—Zvengrowski, P.: On the K-theory and parallelizability of projective Stiefel manifolds, Bol. Soc. Mat. Mex. 31(1) (1986), 29–46.Search in Google Scholar
[2] Barufatti, N. E.—Hacon, D.: K-theory of projective Stiefel manifolds, Trans. Amer. Math. Soc. 352(7) (2000), 3189–3209.10.1090/S0002-9947-00-02614-3Search in Google Scholar
[3] Basu, S.—Chakraborty, P.: On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes, J. Homotopy Relat. Struct. 15(1) (2020), 27–60.10.1007/s40062-019-00244-1Search in Google Scholar
[4] Cartan, H.—Eilenberg, S.: Homological Algebra, Princeton University Press, Princeton, 1956.10.1515/9781400883844Search in Google Scholar
[5] Hodgkin, L. H.—Snaith, V. P.: Topics in K-Theory. The Equivariant K¨unneth Theorem in K-theory. Dyer-Lashof operations in K-Theory. Lecture Notes in Math., Vol. 496, Springer-Verlag, New York, 1975.Search in Google Scholar
[6] He, C.: Cohomology rings of the real and oriented partial flag manifolds, Topology Appl. 279 (2020), Art. ID 107239.10.1016/j.topol.2020.107239Search in Google Scholar
[7] Husemoller, D.: Fibre Bundles. Grad. Texts in Math., Vol. 20, Springer-Verlag, New York, 1975.Search in Google Scholar
[8] Korbaš, J.—Rusin, T.: A note on the Z2-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo (2) 65(3) (2016), 507–517.10.1007/s12215-016-0249-7Search in Google Scholar
[9] MacLane, S.: Homology, Grundlehren Math. Wiss., Vol 114, Springer-Verlag, Berlin, 1967.Search in Google Scholar
[10] Minami, H.: K-groups of symmetric spaces I, Osaka J. Math. 12 (1975), 623–634.Search in Google Scholar
[11] Pittie, H. : Homogeneous vector bundles on homogeneous spaces, Topology 11(2) (1972), 199–203.10.1016/0040-9383(72)90007-9Search in Google Scholar
[12] Podder, S.—Sankaran, P.: K-theory of real Grassmann manifolds, Homology Homotopy Appl. 25(1) (2023), 401–419.10.4310/HHA.2023.v25.n2.a17Search in Google Scholar
[13] Roux, A.: Application de la suite spectrale d’Hodgkin au calcul de la K-theorie des varietes de Stiefel, Bull. Soc. Math. France 99 (1971), 345–368.10.24033/bsmf.1726Search in Google Scholar
[14] Rusin, T.: A note on the cohomology ring of the oriented Grassmann manifolds
[15] Sankaran, P.—Zvengrowski, P.: Stable parallelizability of partially oriented flag manifolds, Pacific J. Math. 128(2) (1987), 349–359.10.2140/pjm.1987.128.349Search in Google Scholar
[16] Sankaran, P.—Zvengrowski, P.: K-theory of oriented Grassmann manifolds, Math. Slovaca 47(3) (1997), 319–338.Search in Google Scholar
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Articles in the same Issue
- Weak differences, weak BCK-algebras and applications to some partial orders on rings
- Weakly κ-compact topological spaces
- On sharp radius estimates for S*(β) and a product function
- Maximal subextension and stability in m-capacity of maximal subextension of m-subharmonic functions with given boundary values
- Asymptotic behavior of fractional super-linear differential equations
- New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform
- Global dynamics of the system of difference equations
- Results on oscillatory properties of third-order functional difference equations with semi-canonical operators
- A new approach to metrical fixed point theorems
- Generalized Baker’s result and stability of functional equations using fixed point results
- Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
- K-theory of oriented flag manifolds
- On certain observations on split continuity and cauchy split continuity
- On the generalized eta- and theta-transformation formulas as the Hecke modular relation
- On some selective star Lindelöf-type properties
