Math Problems Vault

Problem 3.9.1

Prove Proposition 3.9.3: $\pi_1(W_k, x) \simeq F_k$ , where $W_k$ is the wedge (one-point union) of $k$ copies of $S^1$ and $F_k$ is the free group on $k$ letters.

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Remark.

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For $k=2$ : let $x$ be the wedge point of $W_2 = S_1^1 \vee S_2^1$ . Decompose $W_2 = A \cup B$ , where $A$ is the union of $S_1^1$ and a small arc about $x$ in $S_2^1$ , and $B$ is the union of $S_2^1$ and a small arc about $x$ in $S_1^1$ . Then $A \cap B$ deformation-retracts to $x$ , so by the Seifert--van Kampen theorem (in the special case where $\pi_1(A \cap B, x)$ is trivial, $\pi_1(X,x)$ is the free product $\pi_1(A,x) * \pi_1(B,x)$ ), and since $A$ deformation-retracts to $S_1^1$ and $B$ to $S_2^1$ , $\pi_1(W_2, x) \simeq \pi_1(S_1^1, x) * \pi_1(S_2^1, x) \simeq F_1 * F_1 = F_2$ . Adapt this argument for general $k$ .

Problem 3.9.2

(a)(b)(c)

Apply the Seifert--van Kampen theorem to compute the fundamental group of the following spaces:

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(a)

$S^1 \vee S^2$ ;

(b)

$S^n$ , $n \geq 3$ ;

(c)

$S^1 \vee S^n$ , $n \geq 3$ .

Problem 3.9.3

(a)(b)(c)(d)

Consider the dunce hat $D$ , which is made from a triangle by making identifications of its three edges via the pattern $aa^{-1}a^{-1}$ as indicated below.

Dunce hat.

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(a)

For a point $x$ in the interior of the triangle, compute $\pi_1(D \setminus \left\{ x\right\} , y)$ , where $y$ is also in the interior of the triangle.

(b)

Using the decomposition of $D$ with $A = D \setminus \left\{ x\right\}$ , $B =$ interior of triangle, compute $\pi_1(D, y)$ .

(c)

For a standard neighborhood $N$ of a point $x$ on the edge of the triangle, compute $\pi_1(N \setminus \left\{ x\right\} , y)$ . Note that $N$ is the union of three standard half-disk neighborhoods of the three points that are being identified where the small edge segments near those points are also being identified.

(d)

For a vertex $x$ of the triangle (note all three vertices are being identified to one point) and a standard neighborhood $N$ , compute $\pi_1(N \setminus \left\{ x\right\} , y)$ . Here $N$ is formed from three wedges of disk neighborhoods with certain identifications.

Problem 3.9.4

(a)(b)

The projective plane can be considered as the quotient of the disk, where we identify $x \sim -x$ for $x \in S^1$ . The pseudoprojective plane $P_k$ is the quotient of the disk, where we identify $x \sim e^{2\pi i /k} x$ for $x \in S^1$ .

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(a)

Show that $S^1 / x \sim e^{2\pi i/k} x$ is homeomorphic to $S^1$ .

(b)

Use the Seifert--van Kampen theorem to compute $\pi_1(P_k, x)$ .

Remark.

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For part (b): use $A$ to be the complement of the center of the disk in $P_k$ and $B$ to be the interior of the disk.

Problem 3.9.5

Consider the torus $T = S^1 \times S^1 = A \cup B$ , where $A = S^1 \times C \cup \left\{ 1\right\} \times S^1$ and $B = S^1 \times D \cup \left\{ 1\right\} \times S^1$ . Here $C = \left\{ x \in S^1 : x_1 \leq \frac{1}{2}\right\}$ , $D = \left\{ x \in S^1 : x_1 \geq -\frac{1}{2}\right\}$ . Use the Seifert--van Kampen theorem to compute $\pi_1(T, (1,0))$ .

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Remark.

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In computing the induced maps, see where generators are sent geometrically.

Problem 3.9.6

Compute $\pi_1^{ab}(M)$ for the surfaces in the figure below, and use your results and the number of boundary components, to classify the surfaces.

Surfaces for Exercise 3.9.6.

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Problem 3.9.7

Compute $\pi_1^{ab}(M)$ for the surfaces in the figure below, and use your results and the number of boundary components, to classify the surfaces.

Surfaces for Exercise 3.9.7.

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