The Fundamental Group

1 (Spring ’15) #topology/qual/completed

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2 (Spring ’09/Spring ’07/Fall ’07/Fall ’06) #topology/qual/work

• State van Kampen’s theorem.

• Calculate the fundamental group of the space obtained by taking two copies of the torus $T = S^1 \times S^1$ and gluing them along a circle $S^1 \times {p}$ where $p$ is a point in $S^1$.

• Calculate the fundamental group of the Klein bottle.

• Calculate the fundamental group of the one-point union of $S^1 \times S^1$ and $S^1$.

• Calculate the fundamental group of the one-point union of $S^1 \times S^1$ and ${\mathbf{RP}}^2$.

Note: multiple appearances!!

3 (Fall ’18) #topology/qual/work

Prove the following portion of van Kampen’s theorem. If $X = A\cup B$ and $A$, $B$, and $A \cap B$ are nonempty and path connected with ${\operatorname{pt}}\in A \cap B$, then there is a surjection $$\begin{align*} \pi_1 (A, {\operatorname{pt}}) \ast \pi_1 (B, {\operatorname{pt}}) \to \pi_1 (X, {\operatorname{pt}}) .\end{align*}$$

4 (Spring ’15) #topology/qual/work

Let $X$ denote the quotient space formed from the sphere $S^2$ by identifying two distinct points.

Compute the fundamental group and the homology groups of $X$.

5 (Spring ’06) #topology/qual/work

Start with the unit disk ${\mathbb{D}}^2$ and identify points on the boundary if their angles, thought of in polar coordinates, differ a multiple of $\pi/2$.

Let $X$ be the resulting space. Use van Kampen’s theorem to compute $\pi_1 (X, \ast)$.

6 (Spring ’08) #topology/qual/work

Let $L$ be the union of the $z$-axis and the unit circle in the $xy{\hbox{-}}$plane. Compute $\pi_1 ({\mathbf{R}}^3 \backslash L, \ast)$.

7 (Fall ’16) #topology/qual/work

Let $A$ be the union of the unit sphere in ${\mathbf{R}}^3$ and the interval $\left\{{(t, 0, 0) : -1 \leq t \leq 1}\right\} \subset {\mathbf{R}}^3$.

Compute $\pi_1 (A)$ and give an explicit description of the universal cover of $X$.

8 (Spring ’13) #topology/qual/work

• Let $S_1$ and $S_2$ be disjoint surfaces. Give the definition of their connected sum $S^1 #S^2$.

• Compute the fundamental group of the connected sum of the projective plane and the two-torus.

9 (Fall ’15) #topology/qual/work

Compute the fundamental group, using any technique you like, of ${\mathbf{RP}}^2 #{\mathbf{RP}}^2 #{\mathbf{RP}}^2$.

10 (Fall ’11) #topology/qual/work

Let $$\begin{align*} V = {\mathbb{D}}^2 \times S^1 = \left\{{ (z, e^{it}) {~\mathrel{\Big\vert}~}{\left\lVert {z} \right\rVert} \leq 1,~~ 0 \leq t < 2\pi}\right\} \end{align*}$$ be the “solid torus” with boundary given by the torus $T = S^1 \times S^1$ .

For $n \in {\mathbf{Z}}$ define

\begin{align*}
\phi_n : T &\to T \\
(e^{is} , e^{it} ) &\mapsto (e^{is} , e^{i(ns+t)})
.\end{align*}

Find the fundamental group of the identification space $$\begin{align*} V_n = {V{\textstyle\coprod}V \over \sim n} \end{align*}$$ where the equivalence relation $\sim_n$ identifies a point $x$ on the boundary $T$ of the first copy of $V$ with the point $\phi_n (x)$ on the boundary of the second copy of $V$.

11 (Fall ’16) #topology/qual/work

Let $S_k$ be the space obtained by removing $k$ disjoint open disks from the sphere $S^2$. Form $X_k$ by gluing $k$ Möbius bands onto $S_k$ , one for each circle boundary component of $S_k$ (by identifying the boundary circle of a Möbius band homeomorphically with a given boundary component circle).

Use van Kampen’s theorem to calculate $\pi_1 (X_k)$ for each $k > 0$ and identify $X_k$ in terms of the classification of surfaces.

12 (Spring ’13) #topology/qual/work

• Let $A$ be a subspace of a topological space $X$. Define what it means for $A$ to be a deformation retract of $X$.

• Consider $X_1$ the “planar figure eight” and $$\begin{align*}X_2 = S^1 \cup ({0} \times [-1, 1])\end{align*}$$ (the “theta space”). Show that $X_1$ and $X_2$ have isomorphic fundamental groups.

• Prove that the fundamental group of $X_2$ is a free group on two generators.

Spring 2021 #4