12.06主要内容

引子

1. 面临的物理问题 AB effect: \(\oint \vec{A} \cdot d\vec{l}\)
   Monopole: \(\oint \vec{B} \cdot d\vec{S}\); charge: \(\oint \vec{E} \cdot d\vec{S}\)
   ......抽象成Topo term
2. Stokes Theorem(龚昇, \(dxdy \rightarrow dx \wedge dy\))
   \( \int_{c} \mathrm{d} \omega=\int_{\partial c} \omega,\ \text{Or}\ \langle c, \mathrm{d} \omega\rangle=\langle\partial c, \omega\rangle \)
   (1) \(\partial \Sigma_1 = \partial \Sigma_2\)时不影响结果, 可以相差一个边界\(\Sigma_1 - \Sigma_2 = \partial \sigma\),
        等价类\([\Sigma] = \{\Sigma + \partial \sigma | \sigma\}\) \(\Rightarrow\) \(\partial \Sigma \rightarrow \partial \Sigma + \partial^2 \sigma =\partial \Sigma \)不变
   (2) 等价类\([\omega] = \{\omega + d \eta | \eta\}\) \(\Rightarrow\) \(d \omega \rightarrow d \omega + d^2 \eta = d \omega \)不变

同伦群(Homotopy group)

1. 定义: \(f: M\rightarrow N\)之间的mapping等价类形成群\(\pi(X,Y)\), 更常用地\(\pi(S^n,Y)=\pi_n(Y)\)

   

2. 性质:
   (1)\(\pi_n(X \times Y) = \pi_n(X) \times \pi_n(Y) = \pi_n(X) \oplus \pi_n(Y)\);
   (2)\(\pi_n(S^n)=\mathbb{Z}\);
   (3)Table(待补充)

(上)同调群(Cohomology & Homolopy group)

1. 同调群
   (1) 三角剖分; (2) 单形 & 复形; (3) 边界算子\(\partial\): \(\partial^2 = 0\)(建立同调论的基础)
2. 上同调群



3. de Rham同调 & Stokes定理
4. 性质: (1)\(H^r(X \times Y) = \sum_{p+q=r} H^p(X) \times H^q(Y)\)(Kunneth formula); (2)Hurewicz同构定理, 特别地\(H_k(S^n)=\pi_k(S^n)\) if \(k\leq n\); (3)de Rham上同调: \(H_r(M) \simeq H^{n-r}(M)\)

具体应用

1. Winding number
2. Chern number
3. 映射度 & pull back: \({\rm deg}(f)\cdot \int_N \omega = \int_M f^{*} \omega\)

   

总结

USTC| BBS