When I am reading the first chapter of "A Primer on Mapping Class Groups" by Farb and Margalit, there is a beautiful analogy between surfaces and vector spaces. I have interpreted as the following and please correct me if anything is wrong.
| Surface S | Vector Space V |
|---|---|
| Simple closed curves $\alpha$ in $S$ | Vector $v\in V$ |
| Set of essential simple closed curves in S | Basis $\beta$ of $V$ |
| Mod(S)=Homeo$^+$(S,$\partial$ S)/homotopy | GL(V) = GL(n,F)/basis |
| f=[$\phi$]$\in$ Mod(S) | $T=[M]\in GL(V)$, $M$ denoted as some matrix representation of T |
| To understand $f$, we look at $f(\alpha)$ | To understand $T$, we look at $T(v)$ |
| Geometric intersection number i(a,b) | Inner product $\langle a,b\rangle$ |
| Minimal position | ? |
| Bigon | ? |
| Geodesic representation | ? |
However, I have trouble figuring out the last three lines. Moreover, are there more analogy between them?