Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a(G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g(G), and let 1≤r≤2Δ be an integer. In this paper, it is shown that there exists a constant c > 0 such that if g(G)≥cΔ r log(Δ2/r) then χa(G)≤Δ + r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that χ’a(G) = Δ if Δ≥4 and g(G)≥4; or Δ≥3 and g(G)≥5.     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S~2×S~1 or RP~3#RP~3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni(2014) in an earlier paper.     

算子的酉等价与换位代数的K0-群

We show that two irreducible operators on Н are unitarily equivalent if and only if W^*（AФB)‘≌M2(C), and give an answer to the open question posed by J. B. Conway (Subnormal Operators, 7r Pitman, Advanced Publishing Program,Boston, London, Melbourne, 1981) for irreducible operator. We also show that if T, T1 and T2 are irreducible operators with TФT1≌TФT2, then T1≌T2. Finally, we show that K0(A(D))≌Z, giving a new result on the Ko-group of Banach algebras.     

Non-low_2-ness and Computable Lipschitz Reducibility

In this paper, we prove that if a c.e. Turing degree d is non-low_2, then there are two left-c.e. reals β_0, β_1 in d, such that, if β_0 is wtt-reducible to a left-c.e. real α, then β_1 is not computable Lipschitz(cl-) reducible to α. As a corollary, d contains a left-c.e. real which is not cl-reducible to any complex(wtt-complete) left-c.e. real.     

Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces.More precisely, it is shown that(1) if(M, ω) admits a Hamiltonian S~1-action, then there exists a two-sphere S in M with positive symplectic area satisfying c1(M, ω), [S] 0,and(2) if the action is non-Hamiltonian, then there exists an S~1-invariant symplectic2-torus T in(M, ω) such that c1(M, ω), [T] = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott,Lupton-Oprea, and Ono: Suppose that(M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ· [ω] for some λ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then(1) if λ 0, then G must be trivial,(2) if λ = 0, then the G-action is non-Hamiltonian, and(3) if λ 0, then the G-action is Hamiltonian.     

On Reducible Dehn Surgery （ Ⅱ ）

Let K be a knot in a 3-sphere S^3 and N( K ) be a regular neighbourhood of K in S^3. Let Mk = S^3 - intN (K), and T = э Mk. A slope r in T is a T-isotopy class of essential, unoriented, simple, closed curves in T. The distance between two slopes r1 and r2 in T,     

NEW SPECTRAL CHARACTERIZATIONS OF EXTREMAL HYPERSURFACES

Let M be a closed extremal hypersurface in Sn+1 with the same mean curva-ture of the Willmore torus Wm,n?m. We proved that if Specp(M)=Specp(Wm,n?m) for p=0, 1, 2, then M is Wm,m.     

On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere

Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constant γ(n) depending only on n such that if |H| ≤γ(n) and β(n,H)≤S ≤β(n,H)+n/18,then S≡β(n,H) and M is a Clifford torus.Here,β(n,H)=n+n~3/2(n-1)H~2+n(n-2)/2(n-1)(1/2)n~2H~4+4(n-1)H~2.     

Conflict-free Connection Number and Independence Number of a Graph

An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path,which contains a color used on exactly one of its edges.The conflict-free connection number of a connected graph G,denoted by cf c(G),is defined as the minimum number of colors that are required in order to make G conflict-free connected.In this paper,we investigate the relation between the conflict-free connection number and the independence number of a graph.We firstly show that cf c(G)≤α(G)for any connected graph G,and give an example to show that the bound is sharp.With this result,we prove that if T is a tree with?(T)≥(α(T)+2)/2,then cf c(T)=?(T).     

Geometric and topological rigidity for compact submanifolds of odd dimension

We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn(n 5)is an odd-dimensional compact submanifold with parallel mean curvature in Sn+p,and if RicM(n-2-1n)(1+H2)and Hδn,whereδn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn(n 5)is an odd-dimensional compact submanifold in the space form Fn+p(c)with c 0,and if RicM(n-2-εn)(c+H2),whereεn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.     

可约的Dehn手术

本文将利用C.Gordon和J.Luecke发展起来的组合拓扑的方法来证明缆式猜想对一类特殊纽结而言是正确的。     

Annular Dehn Surgery on Knots in a Solid Torus

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Annular Dehn fillings

We show that if a simple 3-manifold M has two Dehn fillings at distance , each of which contains an essential annulus, then M is one of three specific 2-component link exteriors in S ³ . One of these has such a pair of annular fillings with , and the other two have pairs with . Received: February 20, 1999.     

Max Dehn and geometry

Max Dehn is famous for his contributions to group theory and topology, but almost half of his publications are on the foundations of geometry. In this article I hope to show that Dehn's interest in geometry was not misplaced, and that his contributions to it are also remarkable. First, let us survey Dehn's main contributions to geometry, in chronological order.     

On quadratic Dehn functions

We confirm with new examples that Solvable groups of high -rank are expected to satisfy a polynomial isoperimetric inequality ([Gro93] 5A9). To that end we study invariant quasi-geodesic foliations in simply connected solvable Lie groups, endowed with left-invariant Riemannian metrics, whose leaves are isometric to closed subgroups. We establish a decomposition theorem which implies upper bounds on the Dehn (or filling) function (of loops by disks) of the solvable group in terms of the Dehn functions of the leaves. We obtain examples of metabelian polycyclic groups with exponential growth and quadratic Dehn functions. We also deduce that the horospheres in SL(4,)/SO(4,) which bound an invariant core for SL(4, ) and that the horospheres which bound an invariant core for Hilbert modular groups in have quadratic filling functions. The main theorem also applies to some solvable Lie groups which are not quasi-isometric to horospheres in symmetric spaces.Mathematics Subject Classification (2000): 20F65, 20F69, 22E15, 22E40, 53C35Délégation CNRS, UMR 5580     

Roots of Dehn twists

Margalit and Schleimer found examples of roots of the Dehn twist t _C about a nonseparating curve C in a closed orientable surface, that is, homeomorphisms h such that h ⁿ   =  t _C in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the n for which an n ^th root of t _C exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, n ^th roots exist for all genera greater than (n − 2)(n − 1)/2. We also describe all n ^th roots having n greater than or equal to the genus.