maths.free › Algebraic Topology › Homology and the Euler characteristic

Homology and the Euler characteristic

Counting holes in every dimension with chain complexes.

Homology groups H₀, H₁, H₂ count connected pieces, tunnels and voids; their alternating ranks give the Euler characteristic, the same V − E + F from graph theory. Picture it: the cube: 8 − 12 + 6 = 2, like the sphere it can be inflated into. Think it: the boundary of a boundary is zero, ∂² = 0, and that one identity generates the whole theory.

工作范例 · 6 - 12 + 8

Arithmetic · Arithmetic

Evaluate 6 - 12 + 8

6 - 12 + 8

一步

  1. -12 + 6 + 8 = 2

    Add: -12 + 6 + 8 = 2.

发送答案
2

现在轮到你 选择问题, 输入或绘制您自己的 。 每一步, 图片, 答案隐藏到您询问 。

保持你自己的工作

免费账户在每一课上都添加注解, 记录你完成的, 在一个地方解决的, 以及您可以询问的关于此页的导师。 数学本身向每个人开放, 是否签名 。

签名上签名 登录

此处所用符号

输入完整定义、图片和其中每个字母的含义的符号。

如何: Homology and the Euler characteristic

  1. Add: -12 + 6 + 8 = 2.

人们对人们提出的问题

What does the fundamental group measure?

Loops in a space up to deformation. In a plane every loop shrinks to a point (trivial group); around a hole a loop can wind n times (the integers).

更多 Algebraic Topology

The fundamental group