The paper proves group actions on spheres with odd fixed points.
problem Finite group actions on homology six-spheres with odd Euler characteristics.
method Analyzes smooth actions and fixed point sets of finite groups.
result The group is one of three specific types, and the fixed point set is a single point.
New bounds on Seifert surfaces for alternating links are found.
problem Finding the number of Seifert surfaces of fixed genus for alternating links.
method Explicitly given polynomial bound for genus-g Seifert surfaces of fixed Euler characteristic.
result The number of genus-g Seifert surfaces is bounded by a polynomial in the number of crossings.
Affirmatively answers Kosniowski conjecture for unitary S^1-manifolds.
problem Proving Kosniowski conjecture for specific types of manifolds.
method Analyzing unitary S^1-manifolds with isolated fixed points.
result 4χ(M) > dim(M) for the given manifolds, affirming the conjecture.
Polynomial bound on surfaces in hyperbolic 3-manifolds.
problem Bounding the number of surfaces in hyperbolic 3-manifolds.
method Using polynomial functions of the volume of the manifold and the Euler characteristic.
result An upper bound for the number of compact essential surfaces is a polynomial function of the volume of the manifold.
We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for …
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.
This paper studies fixed points of graph selfmaps and iwip endomorphisms of free groups.
problem Fixed points and fixed subgroups of selfmaps on graphs and surfaces.
method Analyzes iwip outer endomorphisms of free groups acting on stable trees.
result A sufficient condition for equality in a bound on a fixed point quantity is found.
A little complement concerning the dynamics of non-metric manifolds is provided, by showing that any flow on an ω-bounded surface with non-zero Euler character has a fixed point.
Study on flexibility of metric entropy and geodesic flow constraints on surfaces.
problem Flexibility and constraints of metrics and flows on surfaces.
method Analysis of geodesic flow, entropy, and metrics in a fixed conformal class.
result Additional restrictions on topological entropy and new geometric properties.
We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also show that when the genus is fixed and the number of punctures varies the behavio…
Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…
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.
Derives log-corrections in AdS4/CFT3 using supergravity localization.
problem Factorizing log-corrections in AdS4/CFT3.
method Supergravity localization, Atiyah-Singer index theorem, fixed points (NUTs), fixed two-manifolds (Bolts).
result General fixed-point formula for log-corrections in large N expansion.
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.
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
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 show that every real analytic action of a connected supersoluble Lie group on a compact surface with nonzero Euler characteristic has a fixed point. This implies that E. Lima's fixed point free C∞ action on S2 of the affine group of the line cannot be approximated by analytic actions. An example is give…
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…
We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as gr…
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.
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…
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.
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.
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…
This paper classifies regular maps with Euler characteristic -p^4 for a prime p≥5.
problem Classify regular maps on surfaces with Euler characteristic -p^4.
method Use inductive method and properties of Sylow p-subgroups to classify.
result Closed surfaces with Euler characteristic -p^4 support no regular maps if p∉{2,3,5,7,13}.
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…
We prove that the minimal Euler characteristic of a closed symplectic four-manifold with given fundamental group is often much larger than the minimal Euler characteristic of almost complex closed four-manifolds with the same fundamental group. In fact, the difference between the two is arbitrarily large for certain gr…
We determine the extent to which the collection of Γ-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of Γ-Euler-Satake characteristics corresponding to free or free abelian Γ and are not…
We prove a surgery formula for the renormalized Euler characteristic of Ozsvath and Szabo. Equality between this Euler cahracteristic and the Seiberg-Witten invariant follows for rational homology three-spheres.
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…
The study bounds the excess of disjoint nonorientable surfaces in a 4-manifold.
problem Bounding the excess of disjoint nonorientable surfaces in a 4-manifold.
method Combining tubing construction with signature and Euler-characteristic formulas for 2-fold branched covers.
result The normal-Euler excess is bounded by a constant depending only on the ambient 4-manifold.
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.
If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique inv…
The Euler characteristic of transitive Lie algebroids vanishes unless they are tangent bundles.
problem Calculating the Euler characteristic of transitive Lie algebroids.
method Atiyah-Singer index theorem and tensor products of elliptic complexes.
result The Euler characteristic of a transitive Lie algebroid vanishes unless it is the tangent bundle.
The study finds infinitely many Lefschetz pencils on ruled surfaces with negative Euler characteristic.
problem Finding Lefschetz pencils on ruled surfaces with negative Euler characteristic.
method Building Lefschetz fibrations on a blow-up of the surface via partial conjugation, then showing compatibility with a symplectic form.
result Ruled surfaces with negative Euler characteristic admit infinitely many inequivalent Lefschetz pencils and fibrations.
Study of maps with -2 Euler characteristic and up to 12 vertices.
problem Enumerating and classifying semi-equivelar maps with specific Euler characteristics.
method Comprehensive enumeration and classification of semi-equivelar maps on a surface with χ=-2, up to 12 vertices.
result Determination of maps' vertex-transitivity and non-transitivity.
Scaling and 1-form addition make projectively equivalent Finsler metrics.
problem Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic.
method Proof of equivalence based on scaling and addition of closed 1-forms.
result Two metrics are projectively equivalent if and only if they differ by a scaling constant and addition of a closed 1-form.
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.
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.