The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
We classify those manifolds of positive euler characteristic on which a lie group G acts with cohomogeneity one, where G is classical simple
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
We fully classify all Lagrangian submanifolds of a complex Grassmannian which are an orbit of a compact group of isometries and have positive Euler characteristic.
Constructs 4-manifolds with positive Euler characteristic proving a conjecture.
problem Finding aspherical 4-manifolds with positive Euler characteristic.
method Explicit construction of 4-manifolds with specified properties.
result Proves a conjecture by constructing a 4-manifold with Euler characteristic 1.
New proof shows 4-manifolds can't support complex structures.
problem Proving 4-manifolds can't support complex structures.
method New proof of complex structure non-existence for real-hyperbolic 4-manifolds.
result Extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure.
We give lower bounds, in terms of the Euler characteristic, for the L2-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.
We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the folia…
The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.
problem Investigating spectral properties of the Jacobi operator for surfaces with nonpositive Euler characteristic.
method Proving a sharp upper bound for the second eigenvalue of the Jacobi operator and classifying surfaces attaining this bound.
result Totally geodesic tori maximize the second eigenvalue among compact orientable surfaces with positive genus.
Study finds symplectic fillings' properties for specific contact covers.
problem Determining symplectic fillings' Euler characteristics and signatures.
method Analyzing exact symplectic fillings of contact branched covers.
result Identified links' symplectic fillings' properties.
The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.
problem Proving positive Euler characteristic for 2d manifolds with specific symmetry groups.
method Direct proof and analysis of fixed point components N with geodesic properties.
result Fixed point components N have amazing geodesic properties and can be S^2, RP^2, CP^d, HP^d, etc.
We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group G with principal isotropy group H and cohomogeneity k such that $k - (\rank G - \ran…
Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…
The paper finds a geometric lower bound for the first positive eigenvalue of the rough Laplacian on 1-forms.
problem Estimating the first positive eigenvalue of the rough Laplacian on 1-forms.
method Establishes a geometric lower bound using assumptions on Ricci curvature, diameter, and Riemann curvature tensor.
result The first positive eigenvalue of the rough Laplacian on 1-forms is bounded below by a positive constant.
The paper extends a conjecture about Euler characteristics of perverse sheaves on certain Kähler manifolds.
problem The conjecture about Euler characteristics of perverse sheaves on compact aspherical Kähler manifolds.
method The method involves expressing the Euler characteristic as an intersection number involving the characteristic cycle, and using curvature conditions to deduce non-negativity. For the second result, the local system is shown to underlie a complex variation of Hodge structures, leading to the desired inequality from curvature properties of the period map.
result The conjecture holds for compact aspherical Kähler manifolds with non-positive holomorphic bisectional curvature and for projective manifolds with a faithful semi-simple rigid local system.
Improved theorem on curvature and manifold symmetry.
problem Closed, even-dimensional manifolds with positive curvature and symmetry.
method Improved four-periodicity theorem and use of characteristic class theory.
result Positive Euler characteristic under specific symmetry conditions.
In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank ≥2 resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic …
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.
The paper proves a stability conjecture for manifolds with zero Euler characteristic.
problem Stability of manifolds with zero Euler characteristic under certain curvature conditions.
method Analyzes manifolds with dimensions 5 or more, proving stability under specific curvature and completeness conditions.
result 2006 Rosenberg's S1-stability holds for manifolds with zero Euler characteristic. The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
New restrictions found on 4-manifolds with pinched curvature.
problem Restrictions on Euler characteristic and signature of 4-manifolds with pinched curvature.
method Proved new restrictions on Euler characteristic and signature of oriented 4-manifolds with pinched sectional curvature.
result Simply connected 4-manifolds with δ≤sec≤1 are homeomorphic to S4 or CP2. This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds i…
Proves Euler characteristic of collapsing Alexandrov spaces.
problem Euler characteristic of collapsing Alexandrov spaces.
method Analyzes strata and fibers of the limit space.
result Euler characteristic equals sum of products of strata and fiber Euler characteristics.
Odd-dimensional orbifolds' Euler characteristic equals half of their boundary's.
problem Calculating the Euler characteristic of odd-dimensional orbifolds.
method Proved through mathematical analysis of orbifolds and their boundaries.
result The Euler characteristic of an odd-dimensional orbifold is half of its boundary's.
Summarizes connections between Euler characteristic theorems and conjectures.
problem Vanishing of the Euler characteristic
method Diagrammatic summary of connections
result Connections between various theorems and conjectures
The Weyl functional is analyzed on 4-manifolds with positive scalar curvature.
problem Analyzing the Weyl functional on 4-manifolds with positive scalar curvature.
method Using integral inequalities and properties of the Weyl tensor.
result Generalized Gursky's inequality for 4-manifolds with positive scalar curvature.
Expands Euler-Poincare characteristic to supergeometry.
problem No new problem introduced.
method Introduced Euler-Poincare characteristic pair in supergeometry.
result Examined transversality in Π-symmetric supermanifolds. We introduce the Γ-Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as t…
Study shows how 3+1D cosmologies can evolve to de Sitter space under certain conditions.
problem Understanding the evolution of 3+1D cosmologies with specific symmetry constraints.
method Mean Curvature Flow methods applied to cosmologies with positive cosmological constant and specific symmetry groups.
result Asymptotically, 3+1D cosmologies evolve to de Sitter space under certain conditions.
The authors give a short survey of previous results on δ-homogeneous Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these resul…
We compute the Euler characteristics of the recently discovered series of Gothic Teichmüller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichmüller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Eul…
For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate n-specialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n≤2, w…
An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open alo…
Formula for manifold Euler characteristic using even faces.
problem Calculating Euler characteristic of triangulated manifolds.
method Formula based on even-dimensional faces.
result Universal coefficients for Euler characteristic.
Proves a generalized table theorem for odd Euler characteristic surfaces.
problem Proving a generalized table theorem for surfaces with odd Euler characteristic.
method Using the square peg problem for smooth curves, the result is generalized to real valued functions on Riemannian surfaces with odd Euler characteristic.
result Proves the table conjecture for even functions on the two sphere.
Revisits and applies a formula for hypersurface Euler characteristics.
problem Understanding isoparametric hypersurfaces in space forms.
method Re-proves Allendoerfer-Weil's formula and applies it to isoparametric hypersurfaces.
result New insights into isoparametric hypersurfaces.
Formula proves Euler characteristic of singularized surfaces.
problem Calculating the Euler characteristic of singularized surfaces.
method Three operations: collapsing, zipping, and double loop identification.
result Formula for Euler characteristic of singularized surfaces.
The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new to…
Formula for Euler characteristic of moduli spaces of Abelian differentials.
problem Computing the Euler characteristic of moduli spaces of Abelian differentials.
method Intersection theory on the smooth compactification by multi-scale differentials, Euler sequence for cotangent bundle, and tools in the Chow ring.
result Formula for the full Chern polynomial of the cotangent bundle.
Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.
problem Addressing the sign of Euler characteristic for manifolds with almost nonnegative curvature operator.
method Analyzing closed manifolds with uniform upper bounds on curvature operator and applying ANCO-type conditions.
result Nonnegative Euler characteristic for closed 2n-dimensional manifolds with almost nonnegative curvature operator and uniform upper bounds on curvature. New Euler characteristic and Burnside group defined for definable groupoids.
problem Defining Euler characteristics and Burnside groups for a broad class of groupoids.
method Introducing universal Euler characteristic and Burnside group for orbit spaces of definable groupoids.
result Every invariant of orbit space definable groupoids arises as a homomorphism of the universal Euler characteristic.
It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes.
In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.
It is well-known that odd-dimensional manifolds have Euler characteristic zero. Furthemore orientable manifolds have an even Euler characteristic unless the dimension is a multiple of 4. We prove here a generalisation of these statements: a k-orientable manifold (or more generally Poincaré complex) has even Euler c…
In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly bounded sectional curvature, then the Euler characteristic χ(M)≥0. Moreover, …
We study a normalized version of the second order renormalization group flow on closed Riemannian surfaces. We discuss some general properties of this flow and establish several basic formulas. In particular, we focus on surfaces with zero and positive Euler characteristic.
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristi…