Euler classes of vector bundles over manifolds
-
Aniruddha C. Naolekar
Abstract
Let đk denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X â đ2n+1 is orientable, then X is a rational homology sphere and Ï1(X) is perfect. We also show that đ8 = â and derive additional cohomlogical restrictions on orientable manifolds in đk.
(Communicated by JĂșlius KorbaĆĄ)
Acknowledgement
We wish to thank Daniel Ruberman and Parameswaran Sankaran for helpful discussions. We wish to acknowledge the detailed comments of the anonymous referee which has resulted in an improved presentation. We are ever indebted to Ganu for his infinite patience and kindness.
References
[1] Bredon, G. E.âKosinski, A.: Vector fields on Ï-manifolds, Ann. of Math. 84(1) (1966), 85â90.10.2307/1970531Suche in Google Scholar
[2] Äadek, M.âVanĆŸura, J.: On oriented vector bundles over CW-complexes of dimension 6 and 7, Comment. Math. Univ. Carolin. 33(4) (1992), 727â736.Suche in Google Scholar
[3] Guijarro, L.âSchick, T.âWalschap, G.: Bundles with spherical Euler class, Pacific J. Math. 27(2) (2002), 377â392.10.2140/pjm.2002.207.377Suche in Google Scholar
[4] Hatcher, A.: Algebraic Topology, https://pi.math.cornell.edu/hatcher/AT/AT.pdfSuche in Google Scholar
[5] Husemoller, D.: Fibre Bundles, Springer-Verlag, New York 1966.10.1007/978-1-4757-4008-0Suche in Google Scholar
[6] Kervaire, M. A.: Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67â7210.1090/S0002-9947-1969-0253347-3Suche in Google Scholar
[7] Mosher, R. E.âTangora, M. C.: Cohomology Operations and Applications in Homotopy Theory, Harper and Row, New York, 1968.Suche in Google Scholar
[8] Naolekar, A. C.âSubhash, B.âThakur, A. S.: On trivialities of Euler classes of oriented bundles over manifolds, Homology Homotopy and Appl. 22(1) (2020), 215â232.10.4310/HHA.2020.v22.n1.a13Suche in Google Scholar
[9] Ruberman, D.: Null-homotopic embedded spheres of codimension one, Tight and Taut submanifolds, MSRI Publications, vol. 32, 1997, pp. 229â232.Suche in Google Scholar
[10] Thomas, E.: Characteristic classes and differentiable manifolds, C.I.M.E. Summer Sch. 41, Springer, Heidelberg, 2010, pp. 113â187.10.1007/978-3-642-11048-1_4Suche in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Prof. RNDr. Ing. LubomĂr KubĂĄÄek, DrSc., Dr.h.c. âNonagenarian
- Doc. RNDr. Roman FriÄ, DrSc. passed away
- Outer and inner approximations in quantum spaces
- Linear derivations on Banach *-algebras
- New fractional order discrete GrĂŒss type inequality
- Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
- Generalized Minkowski type inequality for pseudo-integral
- Study of the Q-spiral-like functions of complex order
- Radius of starlikeness of certain analytic functions
- Successive approximations for a differential equation in a Banach space via Constantin condition
- Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach
- Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Îm
- Sequence selection properties in Cp(X) with the double ideals
- On the paranormed Nörlund difference sequence space of fractional order and geometric properties
- On certain Diophantine equations concerning the area of right triangles
- Weighted projective Ricci curvature in Finsler geometry
- Euler classes of vector bundles over manifolds
- A new generalized Lindley-Weibull class of distributions: Theory, properties and applications
- Dynamical behaviors of a prey-predator model with foraging arena scheme in polluted environments
- The (α, ÎČ)-ramification invariants of a number field
Artikel in diesem Heft
- Regular papers
- Prof. RNDr. Ing. LubomĂr KubĂĄÄek, DrSc., Dr.h.c. âNonagenarian
- Doc. RNDr. Roman FriÄ, DrSc. passed away
- Outer and inner approximations in quantum spaces
- Linear derivations on Banach *-algebras
- New fractional order discrete GrĂŒss type inequality
- Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
- Generalized Minkowski type inequality for pseudo-integral
- Study of the Q-spiral-like functions of complex order
- Radius of starlikeness of certain analytic functions
- Successive approximations for a differential equation in a Banach space via Constantin condition
- Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach
- Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Îm
- Sequence selection properties in Cp(X) with the double ideals
- On the paranormed Nörlund difference sequence space of fractional order and geometric properties
- On certain Diophantine equations concerning the area of right triangles
- Weighted projective Ricci curvature in Finsler geometry
- Euler classes of vector bundles over manifolds
- A new generalized Lindley-Weibull class of distributions: Theory, properties and applications
- Dynamical behaviors of a prey-predator model with foraging arena scheme in polluted environments
- The (α, ÎČ)-ramification invariants of a number field
