# Euler Characteristic - Explore the Science & Experts | ideXlab

## Euler Characteristic

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### Thomas Letendre – One of the best experts on this subject based on the ideXlab platform.

• ##### Expected volume and Euler Characteristic of random submanifolds
Journal of Functional Analysis, 2016
Co-Authors: Thomas Letendre
Abstract:

In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler Characteristic for random submanifolds of codimension $r\in \{1,…,n\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\lambda$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler Characteristic of this random submanifold as $\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler Characteristic of the real vanishing locus of a random real holomorphic section of $\mathcal{E}\otimes\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.

• ##### expected volume and Euler Characteristic of random submanifolds
Journal of Functional Analysis, 2016
Co-Authors: Thomas Letendre
Abstract:

Abstract In a closed manifold of positive dimension n, we estimate the expected volume and Euler Characteristic for random submanifolds of codimension r ∈ { 1 , … , n } in two different settings. On one hand, we consider a closed Riemannian manifold and some positive λ. Then we take r independent random functions in the direct sum of the eigenspaces of the Laplace–Beltrami operator associated to eigenvalues less than λ and consider the random submanifold defined as the common zero set of these r functions. We compute asymptotics for the mean volume and Euler Characteristic of this random submanifold as λ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle L and a rank r holomorphic vector bundle E that are also defined over the reals. Then we get asymptotics for the expected volume and Euler Characteristic of the real vanishing locus of a random real holomorphic section of E ⊗ L d as d goes to infinity. The same techniques apply to both settings.

### Leonardo Macarini – One of the best experts on this subject based on the ideXlab platform.

• ##### On the mean Euler Characteristic of Gorenstein toric contact manifolds
International Mathematics Research Notices, 2018
Co-Authors: Miguel Abreu, Leonardo Macarini
Abstract:

Abstract We prove that the mean Euler Characteristic of a Gorenstein toric contact manifold, that is, a good toric contact manifold with zero 1st Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler Characteristic of a Gorenstein toric contact manifold is equal to the Euler Characteristic of any crepant toric symplectic filling, that is, any toric symplectic filling with zero 1st Chern class.

• ##### On the mean Euler Characteristic of Gorenstein toric contact manifolds
arXiv: Symplectic Geometry, 2016
Co-Authors: Miguel Abreu, Leonardo Macarini
Abstract:

We prove that the mean Euler Characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler Characteristic of a Gorenstein toric contact manifold is equal to the Euler Characteristic of any crepant toric symplectic filling, i.e. any toric symplectic filling with zero first Chern class.

### Alejandro Melle-hernández – One of the best experts on this subject based on the ideXlab platform.

• ##### The universal Euler Characteristic of V-manifolds
arXiv: Geometric Topology, 2018
Co-Authors: Sabir M. Gusein-zade, Ignacio Luengo, Alejandro Melle-hernández
Abstract:

The Euler Characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here we discuss a universal additive topological invariant of V-manifolds: the universal Euler characterictic. It takes values in the ring generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler Characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation”. We give Macdonald type equations for the universal Euler Characteristic for V-manifolds and for cell complexes of the described type.

• ##### The Universal Euler Characteristic of V -Manifolds
Functional Analysis and Its Applications, 2018
Co-Authors: Sabir M. Gusein-zade, Ignacio Luengo, Alejandro Melle-hernández
Abstract:

The Euler Characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler Characteristic. It takes values in the ring freely generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler Characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler Characteristic for V-manifolds and for cell complexes of the described type.

### Primoz Skraba – One of the best experts on this subject based on the ideXlab platform.

• ##### Homological percolation and the Euler Characteristic.
Physical Review E, 2020
Co-Authors: Omer Bobrowski, Primoz Skraba
Abstract:

In this paper we study the connection between the zeros of the expected Euler Characteristic curve and the phenomenon which we refer to as homological percolation-the formation of “giant” cycles in persistent homology, which is intimately related to classical notions of percolation. We perform an experimental study that covers four different models: site percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus T^{d} for d=2,3,4. The simulation results strongly indicate that the zeros of the expected Euler Characteristic curve approximate the critical values for homological percolation. Our results also provide some insight about the approximation error. Further study of this connection could have powerful implications both in the study of percolation theory and in the field of topological data analysis.

### Jacqueline Espina – One of the best experts on this subject based on the ideXlab platform.

• ##### on the mean Euler Characteristic of contact manifolds
International Journal of Mathematics, 2014
Co-Authors: Jacqueline Espina
Abstract:

We express the mean Euler Characteristic (MEC) of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is closed under subcritical contact surgery and examine the behavior of the MEC under such surgery. To this end, we revisit the notion of index-positivity for contact forms. We also obtain an expression for the MEC in the Morse–Bott case.

• ##### On the mean Euler Characteristic of contact manifolds
arXiv: Symplectic Geometry, 2010
Co-Authors: Jacqueline Espina
Abstract:

We express the mean Euler Characteristic of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is closed under subcritical surgery and examine the behavior of the mean Euler Characteristic under such surgery. To this end, we revisit the notion of index-positivity for contact forms. We also obtain an expression for the mean Euler Characteristic in the Morse-Bott case.