38,659 research outputs found
Generalized Orbifold Euler Characteristic of Symmetric Products and Equivariant Morava K-Theory
We introduce the notion of generalized orbifold Euler characteristic
associated to an arbitrary group, and study its properties. We then calculate
generating functions of higher order (p-primary) orbifold Euler characteristic
of symmetric products of a G-manifold M. As a corollary, we obtain a formula
for the number of conjugacy classes of d-tuples of mutually commuting elements
(of order powers of p) in the wreath product G wreath S_n in terms of
corresponding numbers of G. As a topological application, we present generating
functions of Euler characteristic of equivariant Morava K-theories of symmetric
products of a G-manifold M.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-6.abs.htm
Double point self-intersection surfaces of immersions
A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a
double point self-intersection set which is the image of an immersion of a
smooth surface, the double point self-intersection surface. We prove that this
surface may have odd Euler characteristic if and only if k is congruent to 1
modulo 4 or k+1 is a power of 2. This corrects a previously published result by
Andras Szucs.
The method of proof is to evaluate the Stiefel-Whitney numbers of the double
point self-intersection surface. By earier work of the authors these numbers
can be read off from the Hurewicz image h(\alpha ) in H_{2k+2}\Omega ^{\infty
}\Sigma ^{\infty }MO(k) of the element \alpha in \pi _{2k+2}\Omega ^{\infty
}\Sigma ^{\infty }MO(k) corresponding to the immersion under the
Pontrjagin-Thom construction.Comment: 22 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper4.abs.htm
On the dihedral Euler characteristics of Selmer groups of abelian varieties
This note shows how to use the framework of Euler characteristic formulae to
study Selmer groups of abelian varieties in certain dihedral or anticyclotomic
extensions of CM fields via Iwasawa main conjectures, and in particular how to
verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this
setting. When the Selmer group is cotorsion with respect to the associated
Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch
and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with
respect to the associated Iwasawa algebra, we give a conjectural description of
the Euler characteristic of the cotorsion submodule, and explain how to deduce
inequalities from the associated main conjecture divisibilities of Perrin-Riou
and Howard.Comment: 26 pages. Previous discussion of two-variable setting removed, and
discussion of the indefinite setting modified accordingly. To appear in the
HIM "Arithmetic and Geometry" conference proceeding
Local Euler obstructions of toric varieties
We use Matsui and Takeuchi's formula for toric A-discriminants to give
algorithms for computing local Euler obstructions and dual degrees of toric
surfaces and 3-folds. In particular, we consider weighted projective spaces. As
an application we give counterexamples to a conjecture by Matsui and Takeuchi.
As another application we recover the well-known fact that the only defective
normal toric surfaces are cones
- …