时间：2018年1月27日

地点：同济大学宁静楼104教室

上午：自由讨论

下午

主持人：吕志（复旦大学）

1：40-2：30：叶圣奎（西交利物浦大学）

题目：Topological symmetries of simply connected 4-manifolds and Actions of Aut(F_n)

2：40-3：10：陆宇修 （南京大学）

题目：On cobordism of generalized (real) Bott manifolds

主持人：王宏玉（扬州大学）

3：30-4：20：张影（苏州大学）

题目：On the number of topologies on a finite set.

4：30-5：00：吴利苏 (南京大学)

题目：Fundamental groups of small covers.

报告具体摘要

叶圣奎

题目: Topological symmetries of simply connected 4-manifolds and Actions of Aut(F_n)

摘要: Let M be a simply connected closed 4-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on M by homeomorphisms is an abelian group of rank at most two. As applications, let Aut(F_{n}) be the automorphism group of the free group of rank n. It is proved that any group action of Aut(F_{n}) (n≥4) on M≠S⁴ by homologically trivial homeomorphisms factors through Z/2.

陆宇修

题目：On cobordism of generalized (real) Bott manifolds

摘要： We show that all generalized (real) Bott manifolds which are (small covers) quasitoric manifolds over a product of simplices $/Delta^{n_1} /times /cdots /times /Delta^{n_r} /times /Delta^{1}$ are always boundaries of some manifolds. But these manifolds with the natural torus (or $Z_2$-torus) actions do not necessarily bound equivariantly. In addition, we can construct some examples of null-cobordant but not orientedly null-cobordant manifolds among quasitoric manifolds. 张影

题目：On the number of topologies on a finite set

摘要：The numbers T(n) and T_0(n) of distinct topologies and T_0 topologies, respectively, which can be defined on a finite set of n elements exhibit interesting properties. They are actually the numbers of pre-orders and partial orders, respectively, which can be defined on the same set. Furthermore, T(n) can be easily calculated by a combinatorial formula involving T_0(k), k=0, 1, …, n. So far the effort of computer enumeration can evaluate these numbers only for n not exceeding 18. In around 1980 Z. I. Borevich observed periodicity for T_0(n) modulo any positive number m. He proved the result for the case m being a prime, and obtained a consequent periodicity for T(n), while the general case for m being a prime power remains unsolved. In joint work with Xiangfei Li, using ideas of group actions recently introduced by Yasir Kizmaz, we have successfully proved the periodicity for T_0(n) modulo a prime square or cube. Our effort towards proving it modulo arbitrary prime powers and hence arbitrary m is in progress.

吴利苏

题目：Fundamental groups of small covers

摘要： It is well-known that the fundamental group of a small cover is isomorphic to the kernel of a map from the associated Coxeter group to $Z_2^n$, however such description is rough and unpractical. In this talk, I will describe this relation more explicitly based on the presentation of fundamental group we have calculated. Furthermore, some interesting applications will also be introduced. This is joint work with Li Yu.