Euler Characteristic and Topological Phase Transition of NUT-Kerr-Newman Black Hole

From the Gauss-Bonnet-Chern theorem, the Euler characteristic of NUT-Kerr-Newman black hole is calculated to be some discrete numbers from 0 to 2. We find that the Bekenstein-Hawking entropy is the largest entropy in topology by taking into account of the relationship between the entropy and the Euler characteristic. The NUT-Kerr- Newman black hole evolves from the torus-like topological structure to the spherical structure with the changes of mass, angular momentum, electric and NUT charges. In this process, the Euler characteristic and the entropy are changed discontinuously, which give the topological aspect of the first-order phase transition of NUT-Kerr-Newman black hole. The corresponding latent heat of the topological phase transition is also obtained. The estimated latent heat of the black hole evolving from the star just lies in the range of the energy of gamma ray bursts.

MOTIVIC VIRTUAL SIGNED EULER CHARACTERISTICS AND THEIR APPLICATIONS TO VAFA-WITTEN INVARIANTS

For any scheme M with a perfect obstruction theory,Jiang and Thomas associated a scheme N with a symmetric perfect obstruction theory.The scheme N is a cone over M given by the dual of the obstruction sheaf of M,and contains M as its zero section.Locally,N is the critical locus of a regular function.In this note we prove that N is a d-critical scheme in the sense of Joyce.There exists a global motive for N locally given by the motive of the vanishing cycle of the local regular function.We prove a motivic localization formula under the good and circle compact C*-action for N.When taking the Euler characteristic,the weighted Euler characteristic of N weighted by the Behrend function is the signed Euler characteristic of M by motivic method.As applications,using the main theorem we study the motivic generating series of the motivic Vafa-Witten invariants for K3 surfaces.

Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including all those not containing two-sided RP2’s,we determineχ_(4)(π).We address when p(π)equalsχ_(4)(π)and when q^(*)(π)equalsχ_(4)(π),and answer a question raised by Hillman(2002).

Every Graph Embedded on the Surface with Euler Characteristic Numberε=-1 is Acyclically 11-choosable

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G.A graph G is acyclically k-choosable if for any list assignment L={L(v):v∈V(G)}with|L(v)|≥k for each vertex v∈V(G),there exists an acyclic proper vertex coloringφof G such thatφ(v)∈L(v)for each vertex v∈V(G).In this paper,we prove that every graph G embedded on the surface with Euler characteristic numberε=-1 is acyclically 11-choosable.

Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.

Toward a Neurophysiological Measure of Image Quality Perception Based on Algebraic Topology Analysis

The bandwidth of internet connections is still a bottleneck when transmitting large amounts of images,making the image quality assessment essential.Neurophysiological assessment of image quality has highlight advantages for it does not interfere with natural viewing behavior.However,in JPEG compression,the previous study is hard to tell the difference between the electroencephalogram(EEG)evoked by different quality images.In this paper,we propose an EEG analysis approach based on algebraic topology analysis,and the result shows that the difference between Euler characteristics of EEG evoked by different distortion images is striking both in the alpha and beta band.Moreover,we further discuss the relationship between the images and the EEG signals,and the results implied that the algebraic topological properties of images are consistent with that of brain perception,which is possible to give birth to braininspired image compression based on algebraic topological features.In general,an algebraic topologybased approach was proposed in this paper to analyze the perceptual characteristics of image quality,which will be beneficial to provide a reliable score for data compression in the network and improve the network transmission capacity.

Intrinsic Topological Structure of Entropy of Kerr Black Holes

In the light of Ф-mapping method and the relationship between entropy and the Euler characteristic, the intrinsic topological structure of entropy of Kerr black holes is studied. From the Ganss-Bonnet-Chem theorem, it is shown that the entropy of Kerr black hole is determined by singularities of the Killing vector field of spacetime. These singularities naturally carry topological numbers, Hopf indices and Brouwer degrees, which can also be viewed as topological quantization of entropy of Kerr black holes. Specific results S = A/4 for non-extreme Kerr black holes and S = 0 for extreme ones are calculated independently by using the above-mentioned methods.

On the Khovanov Homology of 2- and 3-Strand Braid Links

Although computing the Khovanov homology of links is common in literature, no general formulae have been given for all of them. We give the graded Euler characteristic and the Khovanov homology of the 2-strand braid link ,, and the 3-strand braid .

Symmetric Semi-perfect Obstruction Theory Revisited

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne-Mumford stack X constructed by Chang-Li,and Behrend’s theorem equating the weighted Euler characteristic of X and the virtual count of X by symmetric semi-perfect obstruction theories.As an application,we prove that Joyce’s d-critical scheme admits a symmetric semi-perfect obstruction theory,which can be applied to the virtual Euler characteristics by Jiang-Thomas.

A HOPF INDEX THEOREM FOR A REAL VECTOR BUNDLE

The authors generalize the works in [5] and [6] to prove a Hopf index theorem associated to a smooth section of a real vector bundle with non-isolated zero points.

Variational Problems of Surfaces in a Sphere

Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functional S(x)=∫_(M)S^(n/2)dv,where S is the squared length of the second fundamental form of the immersion x.When x:M→S^(2+p)(1)is a surface in S^(2+p)(1),the functional S(x)=∫_(M)S^(n/2)dv represents double volume of image of Gaussian map.For the critical surface of S(x),we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic.Furthermore,we establish a rigidity theorem for the critical surface of S(x).

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article,we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology,such as small covers,quasi-toric manifolds and(real)moment-angle manifolds;especially for the cases of small covers and quasi-toric manifolds.These kinds of orbit configuration spaces have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope.As a by-product of our method,we also obtain a formula of the Euler characteristic for the classical configuration space,which generalizes the Félix-Thomas formula.In addition,we also study the homotopy type of such orbit configuration spaces.In particular,we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

AN INVARIANT FOR HYPERGRAPHS

The definition for acyclic hypergraphs follows from that for acyclic database schemes. Based on topological structure of Hasse diagram of semilattices, Lee number was proved to be an invariant for hypergraphs, and a necessary and sufficient condition for a hypergraph to be acyclic was given in this paper. Some properties of acyclic hypergraphs were discussed. Some relations for Lee number with several quantities in discrete mathematics were also obtained. We simply discussed some applications of the results in this paper.

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds ∑x（2π- αx） =2αχ（M^2）, where X denotes the Euler characteristic of M^2, αx denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ（τ） as the normed dihedral angle defect around a simplex τ. Our main result is ∑τ（-1）^dim（τ）δ（τ） = χ（K）, where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature k（x） at a vertex and we prove the vertex-version ∑x∈K0 k（x） = χ（K） of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial （n ＋ 1）-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B： χ（W）- 1/2 χ（B） = ∑τ∈W-B（-1）^dim（τ）δ（τ） ＋ ∑τB（-1）^dim（τ）ρ（τ）.

Positive curvature, symmetry, and topology

We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.