拓扑学青年学者论坛

程序册(Program)

=========================================

报告摘要

1. 实moment-angle流形的上同调环

蔡力（中科院数学所）

摘要：一个实moment-angle复形（简记为RMAC）是一个特殊的cubical复形（即每个胞腔是某个维数的立方体），可嵌入（某个维数的）欧式空间的单位立方体之中，并同伦于该欧式空间中一个由坐标平面构成的subspace arrangement的补空（complement）。一个RMAC的拓扑可由其对应的抽象单纯复形K来理解：它是拓扑流形当且仅当K满足一定的组合拓扑条件；其上同调环也可以从由K导出的一个特殊的代数来描述。

2. Self-dual codes and simple polytopes

陈波（华中科技大学）

Abstract: A small cover is a closed n-manifold with a mod 2 torus action and a simple polytope as its orbit space. As known in Toric Topology, many algbraic topologic invaraint could be written in terms of the combinatorics of the simple polytope. The interesting thing in this talk is that the binary self-dual codes in coding theory ouccrs in the equivariant cohomology of some small cover with diagonalsubaction. Furthermore, the codes could be expressed simply and directly by the combinatorics of the simple polytopes.

3. On the character varieties of the pretzel knots

陈海苗（北京工商大学）

Abstract：For a family of the 3-strand pretzel knots, we give a description of their-character varieties, and compute the A-polynomials.

4. Witten Rigidity on Odd Dimensional Manifolds

韩飞（新加坡国立大学）

Abstract: An elliptic operator on a manifold is called rigid if its equivariant index is constant or vanishing when the manifold admits a nontrivial Lie group action. Classical operators like the signature operators, the Dirac operators and etc. are rigid. The Witten rigidity conjecture predicts the rigidity of many important twisted Dirac operators on even dimensional spin manifolds and was proved by Bott, Taubes and Liu. In this talk, I will describe Witten type rigidity theorems on odd dimensional spin manifolds. This represents our joint work with Jianqing Yu.

5. Topology of local symplectic conifold transitions on-bundles

江怡（清华大学数学科学中心）

Abstract: Symplectic conifold transitions are symplectic surgeries along Lagrangian spheres in symplectic 6-manifolds. A natural question is if symplectic conifold transitions preserve Kahler structures. As the projectivizations of rank two holomorphic vector bundles over Kahler surfaces form a typical family of Kahler 3-folds, we prove the local symplectic conifold transitions of these 3-folds are all diffeomorphic to Kahler 3-folds. More generally, we determine the diffeomorphism types of local symplectic conifold transitions on all-bundles over symplectic 4-manifolds.

6．Arithmetic hyperbolic lattices with a fixed invariant trace field

马继明（复旦大学）

Abstract: We study the distribution of arithmetic hyperbolic lattices with a fixed invariant trace field, we show the number of maximal arithmetic lattices inwith covolume at mostis bounded by polynomials on.

7. The fixed point data and equivariant Chern numbers

马骏 (复旦大学)

Abstract： First, we follow the line of Tom Dieck to consider the unitary-equivariant cobordism and prove equivariant Chern numbers determine the bordism class of a stable almost complex-manifold. For given set of trivialaction connected compact unitary manifolds andcomplex vector bundle of them, we find necessary and sufficient conditions for the existence of a stable almost complexmanifold with the given fixed point data by using equivariant Chern numbers .

8. A Bona Fide CW Decomposition of Compact Manifolds Resulting from Morse Theory

秦理真 (南京大学)

摘要：Consider a Morse function on a compact Riemannian manifold. This manifold is a disjoint union of unstable manifolds of the negative gradient of the Morse function. These unstable manifolds are open cells. Does this give a CW complex which is homeomorphic (not only homotopy equivalent) to the compact manifold? This talk will describe the history of the above problem and give an affirmative answer to it.

9. Free involutions on some special manifolds

杨建强 （四川大学）

Abstract: The classification of manifolds with certain properties up to homeomorphism or diffeomorphism is one of the main topics in geometric topology. I classify some smooth 5-manifolds with fundamental group isomorphic toand universal cover diffeomorphic to some special manifolds. This gives a classification of smooth free involutions on these special manifolds up to conjugation. It is an application for surgery and modified surgery theory.

10. Title：TBA.

王诗宬（北京大学）

Abstract: TBA.

11. Generators of some bordism groups

王慰（上海海洋大学）

Abstract: The main purpose of this talk is to find some typical manifolds to represent the generators of some bordism groups geometrically. This talk divides into two parts.

In the first part, we will collect some basic facts about the structures of some bordism groups. In the second part, we will choose some classes of manifolds (highly-connected framed manifolds, quasitoric manifolds, etc.) to represent some generators of these bordism groups.

12. On Minimal Genus Problem

吴忠涛 （香港中文大学）

Abstract: In joint work with Yi Ni, we obtained a lower bound for the rational genus of knots in a given homology class. We discuss its connection to the Z/2-Thurston norm and complexity of manifolds, and speculate a slice genus version of the bound.

13. Rotating families with applications to Mapping class groups.

杨文元 （北京大学）

Abstract: This talk will present Dahmani-Guirardel-Osin’s work on the existence of free, purely pseudo-Anosov, normal subgroups of Mapping class groups. Their tool is the theory of rotating family which was introduced by Gromov.

14. Matrix group actions on CAT(0) spaces and Spheres

叶圣奎 (西交利物浦大学)

Abstract: Letbe any finitely generated ring andan integer. Suppose thatis the matrix group generated by all elementary matrices. We will show that

1) any semisimple isometric action of on a-dimensionalspace has a global fixed point if;

2) any action ofby homeomorphisms on a k-sphere is trivial if.

15. On a class of quotient spaces of moment-angle complexes

于立（南京大学）

Abstract：For a simplicial complexand a partitionof the vertex set of, we define a quotient space of the (real) moment-angle complex ofby some (not necessarily free) torus action determined by. We obtain an analogue of Hochster's formula to compute the cohomology groups of such a space with any coefficients. Moreover, we show that their cohomology rings with-coefficient are isomorphic as multigraded-modules (or algebras) to the cohomology of some multigraded differential algebra determined byand. This generalizes the isomorphism between the cohomology ring of a moment-angle complexand the Tor algebra of the face ring of. In addition, we explain how to extend these results to a wider range of spaces.

References:

[1]. A. Bahri, M. Bendersky, F.Cohen and S.Gitler, The Polyhedral Product Functor: a method of computation for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), 1634--1668.

[2]. V.M. Buchstaber and T.E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, 24. American Mathematical Society, Providence, RI, 2002.

[3]. V.M. Buchstaber and T.E. Panov, Toric topology, arXiv:1210.2368.

[4]. X. Cao and Z.Lu, Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture, J. Algebraic Combin. 35 (2012), 121--140.

[5]. M.W. Davis and T. Januszkiewicz, Convex polytopes,

Coxeter orbifolds and torus actions}, Duke Math. J. 62

(1991), 417--451.

[6]. Z. Lu and T.E. Panov, Moment-angle complexes from simplicial posets, Cent. Eur. J. Math. 9 (2011), 715--730.

[7]. M. Masuda and T.E. Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006), 711--746.

16. Homotopy decompositions in homotopy theory

赵浩 (华南师范大学)

Abstract: Homotopy decompositions play an important role in determining the homotopy properties of a topological space. For example, we can decompose a space as a product or wedge of some indecomposable factors, investigate them and the way they are assembled to understand the original space. In this talk I will introduce the method of homotopy decomposition developed recently and show some of our recent results obtained via homotopy decomposition.

曲面映射类群Mini-course 摘要与相关文献

1. Generators of the mapping class groups

杜晓明（华南理工大学）

摘要：(1) 曲面映射类群有限生成的Lickorish定理

(2) 曲面映射类群由两个元素生成的Korkmaz定理

参考文献

[1] Farb and Margalit, A primer on mapping class groups, Chapter 3-4

[2] Korkmaz, Generating the surface mapping class group by two elements, Transactions of the A.M.S., Volume 357, Number 8, Pages 3299–3310 (2004)

2. Torelli group and Johnson homeomorphism

刘勍 (北京大学)

摘要：关于Torelli group的基本内容以及curve complex的基本内容。包括：

1，mapping class group到辛群的是满同态，由此定义Torellii group。

2，Torelli group作用在curve complex上的轨道。

3，Johnson homomorphism

参考文献：

[1] Benson Farb and Dan Margalit. A primer on mapping class groups.

[2] Dennis Johnson. An abelian quotient of the mapping class group

3. Nielsen realization theorem

王家军（北京大学）

摘要: This will be a short lecture on Kerckhoff's proof of the Nielsen realization theorem.

参考文献：

Steven P Kerckhoff. Nielsen realization theorem

Annals of Mathematics, Second Series, 1983 vol. 117 (2) pp. 235-265.http://dx.doi.org/10.2307/2007076