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National Science Review 2019年第5期6卷 962-969页

作者： Dinghua Shi Linyuan Lü Guanrong Chen Department of Mathematics College of Science Shanghai University Institute of Fundamental and Frontier Sciences University of Electronic Science and Technology of China Department of Electronic Engineering City University of Hong Kong

In network science, the non-homogeneity of node degrees has been a concerning issue for ***, with today’s modern web technologies, the traditional social communication topologies have evolved from node-central structures into online cycle-based communities, urgently requiring new network theories and *** the focus from node degrees to network cycles could reveal many interesting properties from the perspective of totally homogenous networks or sub-networks in a complex network, especially basic simplexes(cliques) such as links and ***, compared with node degrees, it is much more challenging to deal with network *** studying the latter, a new clique vector-space framework is introduced in this paper, where the vector space with a basis consisting of links has a dimension equal to the number of links, that with a basis consisting of triangles has the dimension equal to the number of triangles and so *** two vector spaces are related through a boundary operator, for example mapping the boundary of a triangle in one space to the sum of three links in the other *** the new framework,some important concepts and methodologies from algebraic topology, such as characteristic number,homology group and Betti number, will play a part in network science leading to foreseeable new research *** immediate applications, the paper illustrates some important characteristics affecting the collective behaviors of complex networks, some new cycle-dependent importance indexes of nodes and implications for network synchronization and brain-network analysis.

关键词： boundary operator clique vector space cycle homology group totally homogenous network

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Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application

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Acta Mathematica Sinica,English Series 2018年第11期34卷 1742-1754页

作者： Deng Pin LIU Department of Mathematics Guangxi Normal University

Let Pn be a simple n-polytope with a Z2-characteristic function λ. And h is a Morse function over Pn. Then the small cover Mn（λ） corresponding to the pair （Pn, λ） has a cell structure given by h. From this cell structure we can derive a cellular chain complex of Mn（λ） with integer coefficients. In this paper, firstly, we discuss the highest dimensional boundary morphism ？n of this cellular chain complex and get that ？n=0 or 2 by a natural way. And then, from the well-known result that the submanifold corresponding to （F, λF） is naturally a small cover with dimension k, where F is any k-face of Pn and λF is the restriction of λ on F, we get that ？k=0 or ±2 for 0 ≤ k 〈 n. Finally, by using the definition of inherited characteristic function which is the restriction of λ on the faces of Pn, we get a way to calculate the homology groups of Mn（λ）. Applying our result to a 3-small cover we have that the homology groups of any 3-small cover is torsion-free or has only 2-torsion.

关键词： Small cover homology group orientation CW-complex

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Geometric and topological rigidity for compact submanifolds of odd dimension

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Science China Mathematics 2014年第7期57卷 1525-1538页

作者： XU HongWei LENG Yan GU JuanRu Center of Mathematical Sciences Zhejiang University

We investigate rigidity problems for odd-dimensional compact *** show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2） and H ＆lt; δn,where δn is an explicit positive constant depending only on n,then M is a totally umbilical *** H is the mean curvature of ***,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 ＆gt;（n-2-εn）（c＋H2）,where εn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.

关键词： submanifolds geometric and topological rigidity Ricci curvature stable currents homology group

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Geometry and Topology of Spaces of Quasi-constant Curvature

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Communications in Mathematical Research 1998年第4期 118-125页

作者：张学山

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＮｎｂｅａｎｎ（≥３）－ｄｉｍｅｎｓｉｏｎａｌＲｉｅｍａｎｎｉａｎｍａｎｉｆｏｌｄ，Ｘ∈ＴｘＮａｎｄＣ（Ｘ，θ）＝｛２－ｐｌａｎｅσＴｘＮ；ｔｈｅａｎｇｌｅｂｅｔｗｅｅｎσａｎｄ... 详细信息

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＮｎｂｅａｎｎ（≥３）－ｄｉｍｅｎｓｉｏｎａｌＲｉｅｍａｎｎｉａｎｍａｎｉｆｏｌｄ，Ｘ∈ＴｘＮａｎｄＣ（Ｘ，θ）＝｛２－ｐｌａｎｅσＴｘＮ；ｔｈｅａｎｇｌｅｂｅｔｗｅｅｎσａｎｄＸｉｓθ｝．（１．１）Ｉｎ１９７８...

关键词： Quasi constant Curvature Stable Integral Current homology group

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