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# 数学代写|拓扑学代写TOPOLOGY代考|The Fundamental Group of a Compact Surface

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## 数学代写|拓扑学代写TOPOLOGY代考|The Fundamental Group of a Compact Surface

We’ll start with two examples that show how the second version of the Seifert-Van Kampen Theorem can be used to compute the fundamental group of a compact surface presented as an identification space.

Example Let $\mathbb{T}$ be the torus, presented as the ID space $a b a^{-1} b^{-1}$ obtained by identifying the sides of a rectangle in the usual way. Let $B$ be contained in the interior of the rectangle, consisting of most of the rectangle not quite all the way to its boundary. Let $A$ be the remaining part of the rectangle-extended a bit into the interior of $B$. Then the intersection $A \cap B$ is a “ribbon” that runs parallel to the boundary of the rectangle. See Figure 12.6.

Now $B$ is homotopic to an open disk, which is contractible and thus has trivial fundamental group. Also $A \cap B$ is an annulus, which deformation retracts onto the circle and has fundamental group $\pi_1(A \cap B) \cong \mathbb{Z}$. What about $A$ ? By folding the rectangle up into a cylinder by gluing together the $a$-edge, we can see that $A$ is homotopic to a thickened “figure eight,” which deformation retracts onto the wedge of two circles and has $\pi_1(A) \cong \mathbb{Z} * \mathbb{Z}$. Concretely, we can say that $\pi_1(A)$ is the free group $F([a],[b])$.

In order to apply the Seifert-Van Kampen Theorem, we must know how $\pi_1(A \cap$ $B$ ) injects into $\pi_1(A)$ under the homomorphism induced by the inclusion map $\iota: A \cap B \rightarrow A$. To this end, observe that the curve $\gamma$ pictured in Figure 12.6, whose equivalence class generates $\pi_1(A \cap B)$, is a curve that winds once around the rectangle by following curve segments that are almost-but not quite equal to-the edges of the rectangle. In fact, we can say that $\gamma \sim a * b * \bar{a} * \bar{b}$, where we recall that $$is curve concatenation and the bar denotes reversed orientation. Therefore, \iota_[\gamma]=[a][b][a]^{-1}[b]^{-1} The normal subgroup N of the Seifert-Van Kampen Theorem is therefore the subgroup of F([a],[b]) that contains [a][b][a]^{-1}[b]^{-1} along with all elements generated by all conjugates of [a][b][a]^{-1}[b]^{-1}. Hence, in the quotient group, we’ll have [a][b][a]^{-1}[b]^{-1}=[e] or else [a][b]=[b][a]. The quotient group is abelian! In fact, we’ll find$$
\begin{aligned}
\pi_1(\mathbb{T}) & \cong F([a],[b]) / N \
& \cong\langle[a],[b] \mid[a][b]=[b][a]\rangle \
& \cong \mathbb{Z} \times \mathbb{Z}
\end{aligned}

## MATLAB代写

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