这是一份Sydney悉尼大学MATH3076/MATH3976 的成功案例

Disjoint union axiom): Let $X_{1}$ and $X_{2}$ be compact oriented surfaces with boundaries whose labels are $\ell_{1}$ and $\ell_{2}$ respectively.
$$
F\left(X_{1} \text { I } X_{2}, \ell_{1} \text { Ш } \ell_{2}\right)=F\left(X_{1}, \ell_{1}\right) \otimes F\left(X_{2}, \ell_{2}\right) .
$$
(Gluing axiom): Let $\tilde{X}$ be a compact oriented surface obtained by sluing together a pair of boundary circles with dual labels in $X$.
$$
F(\tilde{X}, \ell)=\bigoplus_{x \in L} F(X, \ell \cup{x, \hat{x}})
$$
Duality axiom): Let $X^{}$ stnds for $X$ with the reversed orientation and labels applied $.$ $$ F\left(X^{}\right)=(F(X))^{} \text {, where we mean that } A^{}={ }^{t} \bar{A} .
$$

MATH3076/MATH3976 COURSE NOTES ：

Let $X$ be a topological space. For any $p, q \in X$ if there is a continuous $\operatorname{map} f:[0,1] \rightarrow X$ such that $f(0)=p, f(1)=q$, then $X$ is called arcwise connected and such a map is called a path in $X$. In what follows, we always consider an arcwise connected topological space and so we refer it simply “a space” hereafter.

Let $X$ be a space, and for $p, q, r \in X$ let $f$ be a path connecting $p$ with $q$ and $g$ a path connecting $q$ with $r$. Then we define a product of paths, $f \cdot g$, as follows:
$$
f \cdot g(s)= \begin{cases}f(2 s) & (0 \leq s \leq 1 / 2) \ g(2 s-1) & (1 / 2 \leq s \leq 1)\end{cases}
$$
The $f \cdot g$ is a new path in $X$ connecting $p$ with $r$.
When there is a path $l$ such that the starting point and the end point coincide, i.e., a continuous map
$$
l:[0,1] \rightarrow X, l(0)=p, l(1)=p
$$
is called a loop in $X$ with a base point $p$.