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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?

quuuuuin
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    The 2nd item on the list is off: It is like saying that a basis is simply a subset of a vector space. You can say "a minimal filling collection of simple loops." Also, GL(n,F)/basis does not make sense. Item 5 is a general triviality: A map/function is determined by its values. Another issue: When studying MCG, you are not working with simple loops, but with isotopy classes of simple loops. There is a way to embed the set of isotopy classes of simple loops in a vector space, where MCG acts linearly and $i(a,b)$ is a bilinear form. But then you have to learn about laminations and currents. – Moishe Kohan May 28 '22 at 13:49
  • @MoisheKohan Thanks so much for your comments! Is minimal filling analogous to spanning in vector space? For the isotopy class of simple loops, is there a way to find an analogy in the vector space e.g. the multiple of a vector $av,a\in F, v\in V$? – quuuuuin May 28 '22 at 14:08
  • These are all only loose analogies. One can say that a filling set of curves is an analogue of a spanning subset in a vector space, since it an element of MCG is determined by its values on members of a filling collection of curves. To get to a basis, you have to impose the minimality condition on a spanning set. As for your last question: Yes, but you are not ready for this. – Moishe Kohan May 28 '22 at 14:12
  • Thanks for the clarification! I will keep the analogies in mind and fill them better as I learn further! – quuuuuin May 28 '22 at 14:24

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