Powers of the Euler Class

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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Euler classes of vector bundles over manifolds

Mathematica Slovaca ◽

10.1515/ms-2017-0461 ◽

2021 ◽

Vol 71 (1) ◽

pp. 199-210

Author(s):

Aniruddha C. Naolekar

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Euler Class ◽

Homology Sphere ◽

Rational Homology ◽

Rational Homology Sphere ◽

Orientable Manifolds

Abstract Let 𝓔 k 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 ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

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Theta functions on T 2-bundles over T 2 with the zero Euler class

Siberian Mathematical Journal ◽

10.1007/s11202-009-0072-x ◽

2009 ◽

Vol 50 (4) ◽

pp. 647-657 ◽

Cited By ~ 2

Author(s):

D. V. Egorov

Keyword(s):

Theta Functions ◽

Euler Class

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On invariance of the Euler class groups under a subintegral base change

Journal of Algebra ◽

10.1016/j.jalgebra.2013.09.024 ◽

2014 ◽

Vol 398 ◽

pp. 131-155 ◽

Cited By ~ 4

Author(s):

Mrinal Kanti Das ◽

Md. Ali Zinna

Keyword(s):

Euler Class ◽

Base Change ◽

Class Groups

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The Euler class of an umbilic foliation

Comptes Rendus Mathematique ◽

10.1016/j.crma.2016.03.014 ◽

2016 ◽

Vol 354 (6) ◽

pp. 614-618 ◽

Cited By ~ 1

Author(s):

Icaro Gonçalves ◽

Fabiano Brito

Keyword(s):

Euler Class

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The Euler class and periodicity of group cohomology

Commentarii Mathematici Helvetici ◽

10.1007/bf02566105 ◽

1978 ◽

Vol 53 (1) ◽

pp. 643-650 ◽

Cited By ~ 8

Author(s):

Stefan Jackowski

Keyword(s):

Euler Class ◽

Group Cohomology

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ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

Bulletin of the London Mathematical Society ◽

10.1112/s002460930100892x ◽

2002 ◽

Vol 34 (3) ◽

pp. 329-340 ◽

Cited By ~ 4

Author(s):

BRAD LACKEY

Keyword(s):

Finsler Space ◽

Finsler Geometry ◽

Euler Class ◽

Constant Volume ◽

Finsler Spaces ◽

Chern Connection ◽

Civita Connection ◽

Levi Civita Connection ◽

Riemannian Case ◽

Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

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A TORSIONAL TOPOLOGICAL INVARIANT

International Journal of Modern Physics A ◽

10.1142/s0217751x07038414 ◽

2007 ◽

Vol 22 (29) ◽

pp. 5237-5244 ◽

Cited By ~ 55

Author(s):

H. T. NIEH

Keyword(s):

Affine Connection ◽

Euler Class ◽

Christoffel Symbol ◽

Space Time ◽

Topological Invariant ◽

Point Of View ◽

Underlying Space ◽

Recent Developments ◽

Theory Point ◽

Curvature And Torsion

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

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