文章与日志

Basic Notions of Manifold

When we mention Euclid space, maybe it induct excess meanings. So,
we can try to construct an object that its meaning is less than
Euclid space but however, is meaningful and analyzable.

At least, we need some points, then, if we want a point is
analyzable, it must has its neighborhood, moreover, to make it
computable or realizable, let its neighborhood homeomorphism with
an open set of Euclid space.

Such points constitutes a set, then make this set is a separated
set, a Hausdorff set, we definite such a set a manifold.

The point `x\in M`, M is a manifold. U is a neighborhood of x, and
there is a homeomorphism `\phi:U\mapsto \phi(U)`. then, Any such a
$(U,\phi(U))$ means a realization of any point of U: $\phi(y)$ is

只是在实数上出现的分析问题，就有可能困难得要求把它置于更一般的基底，才有可能找到解答。
首先有下面两个准备好的东西：

1.拓扑结构
一个满足[[第二可数公理]]的[[Hausdorff空间]]M
2.把实数域扩充为欧式空间R
以集合的手段做成一个实数组[tex]\{x_1,x_2,...x_n\}[/tex],以这样一个数组做成一个元素的样式，再在这样的元素之间定义加法和实数对它的乘法，就架起了一个矢量空间。

何谓分类？
已经找到2维流形的清晰分类：
任何一个闭2维曲面，都可以由3种基本曲面通过离散群作用求商集得到。
这三基本曲面是2维球面，欧氏平面，双曲盘面。
所以分类落实在曲面之间的拓扑变换关系。

那么3维情形呢？

Poincaré还是从球面入手
有理由认为球面具有非常尊贵的几何地位，Poincaré提出如下猜想：
如果一个3维闭流形与3维球面同调，那么它必定与之同胚。
不久他找到一个反例SO(3)/I_60，于是再问：
如果一个3维闭流形具有平凡基本群，那么它是否一定与3维球面同胚？