Math4202 Topology II (Lecture 36)

Algebraic Topology

Classification of surfaces

Recall from previous lecture, let $X=P/\sim$ be the space obtained from gluing the polygonal region by pasting edges together according to same labeling.

Definition of $n$ fold torus

Let $P$ be a $4n$-sideded polygonal region

With $a_1b_1a_1^{-1}b^{-1}a_2b_2a_2^{-1}b_2^{-1}\cdots a_nb_na_n^{-1}b_n^{-1}$, the quotient space associated with the above labelling is called the $n$ fold torus.

Denote by $T\# T\# \cdots \# T$.

Definition of $m$-fold connected sum of projective planes

Let $m>1$ $2m$-sided polygonal region, with $a_1^2a_2^2\cdots a_m^2$, the quotient space is called $m$-fold connected sum of projective planes.

Denote by $P^2\# P^2\# \cdots \# P^2$.

Theorem for the fundamental group of $n$-fold torus

$\pi_1(T\# T\# \cdots \# T,x_0)=\langle a_1,a_2,\cdots,a_n \rangle/N(a_1b_1a_1^{-1}b^{-1}a_2b_2a_2^{-1}b_2^{-1}\cdots a_nb_na_n^{-1}b_n^{-1})$.

And

$\pi_1(P^2\# P^2\# \cdots \# P^2,x_0)=\langle a_1^2,a_2^2,\cdots,a_m^2 \rangle/N(a_1^2a_2^2\cdots a_m^2)$

Abelianization

Recall the commutator subgroup?

Let $G$ be a group, and $G/N=[G,G]$ be the smallest subgroup $g_1g_2g^{-1}g_2^{-1}\cdots g_n^{-1}$.

First homology group of $n$-fold torus

First homology group of $m$-fold connected sum of projective planes

Theorem: That’s all we need as $2$ dimensional surfaces

Note that the labeling have some equivalence relations

1. $y_0y_1\cdots\sim y_0cc^{-1}y_1\cdots$ (add edges)
2. $y_0cc^{-1}y_1\cdots\sim y_0y_1\cdots$ (remove edges)
3. Relabelling should be allowed
4. Cyclic permutation should be allowed
5. Flip should be allowed

Up to all these relations, we end up with the following 3 class of labels

1. $aa^{-1}bb^{-1}$ Sphere
2. $a_1^2a_2^2\cdots a_m^2$ Projective plane
3. $a_1b_1a_1^{-1}b^{-1}a_2b_2a_2^{-1}b_2^{-1}\cdots a_nb_na_n^{-1}b_n^{-1}$ Torus