Euler classes of vector bundles over iterated suspensions of real projective spaces
-
Aniruddha C. Naolekar
and
Ajay Singh Thakur
Abstract
We show that when k ≠ 2, 4, 8 the Euler class of any vector bundle over Σkℝℙm is zero if the rank of the bundle is not m + k, provided that m ≠ 3 when k = 6. If k = 2, 4, 8 we show that the Euler class of any vector bundle over Σkℝℙm is zero whenever the rank of the bundle is not kr + k, provided that m ≠ 6, 7 when k = 2, where r is the largest integer such that kr ≤ m.
The research of second-named author has been supported by Indian Statistical Institute, Bangalore and DST-Inspire Faculty Scheme (IFA-13-MA-26).
Acknowledgement
The authors acknowledge referees of this paper for their suggestions that have made the results stronger and improved the presentation of the paper.
References
[1] Atiyah, M.—Hirzebruch, F.: Bott periodicity and the parallelizability of the spheres, Proc. Cambridge Philos. Soc. 57 (1961), 223–226.10.1017/S0305004100035088Search in Google Scholar
[2] Guijarro, L.—Schick, T.—Walschap, G.: Bundles with spherical Euler class, Pacific J. Math. 27 (2002), 377–392.10.2140/pjm.2002.207.377Search in Google Scholar
[3] Čadek, M.—Vanžura, J.: On the classification of oriented bundles over 9-complexes. In: Proc. of Winter school “Geometry and Physics”, Circolo Matematico di Palermo, Palermo, Rend. Circ. Mat. Palermo (2) Suppl. 37 (1994), pp. 33–40.Search in Google Scholar
[4] Fujii, M.: KO-groups of projective spaces, Osaka J. Math. 4 (1967), 141–149.Search in Google Scholar
[5] Malyĭ, B. D.: The Euler classes of stably equivalent vector bundles, Izv. Akad. SSSR Ser. Mat. 38 (1974), 107–126.10.1070/IM1974v008n01ABEH002098Search in Google Scholar
[6] Milnor, J.—Stasheff, J.: Characteristic Classes, Princeton Univ. Press, Princeton, 1974.10.1515/9781400881826Search in Google Scholar
[7] Naolekar, A. C.: Realizing cohomology classes as Euler classes, Math. Slovaca 62 (2012), 949–966.10.2478/s12175-012-0057-2Search in Google Scholar
[8] Tanaka, R.: On trivialities of Stiefel-Whitney classes of vector bundles over iterated suspension spaces, Homology Homotopy Appl. 12 (2010), 357–366.10.4310/HHA.2010.v12.n1.a18Search in Google Scholar
[9] Walschap, G.: The Euler class as a cohomology generator, Illinois J. Math. 46 (2002), 165–169.10.1215/ijm/1258136147Search in Google Scholar
[10] Woodward, L. M.: The classification of orientable vector bundles over CW-complexes of low dimensions, Proc. Royal Soc. Edinburgh Sect. A 92(3-4) (1982), 175–179.10.1017/S0308210500032467Search in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Small-large subgroups of the reals
- Structural properties of algebras of S-probabilities
- Approximation in quantum measure spaces
- On Fibonomial sums identities with special sign functions: analytically q-calculus approach
- On the proximity of multiplicative functions to the number of distinct prime factors function
- Truncated euler polynomials
- Growth series of crossed and two-sided crossed products of cyclic groups
- Weakly U-abundant semigroups with strong Ehresmann transversals
- Iterative learning control with pulse compensation for fractional differential systems
- Rectifiable and nonrectifiable solution curves of half-linear differential systems
- Uniqueness properties of meromorphic functions in the light of three shared sets
- A new family of analytic functions defined by means of Rodrigues type formula
- Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates
- On the solutions of a second-order difference equation in terms of generalized Padovan sequences
- Fixed points in C∗-algebra valued b-metric spaces endowed with a graph
- A general fixed point theorem for two pairs of mappings satisfying a mixed implicit relation
- Productively sequential spaces
- Euler classes of vector bundles over iterated suspensions of real projective spaces
- The stationary distribution and ergodicity of a stochastic mutualism model
- Restricted injective dimensions over local homomorphisms
