# The Fundamental Group

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

Let $$S^1$$ denote the unit circle in $$C$$, $$X$$ be any topological space, $$x_0 \in X$$, and \begin{align*}\gamma_0, \gamma_1 : S^1 \to X\end{align*} be two continuous maps such that $$\gamma_0 (1) = \gamma_1 (1) = x_0$$.

Prove that $$\gamma_0$$ is homotopic to $$\gamma_1$$ if and only if the elements represented by $$\gamma_0$$ and $$\gamma_1$$ in $$\pi_1 (X, x_0 )$$ are conjugate.

• Any two maps $$f_i: Y\to X$$ are homotopic iff there exists a homotopy $$H: I\times Y \to X$$ with $$H_0 = f_0$$ and $$H_1 = f_1$$.

• $$\pi_1(X; x_0)$$ is the set of maps $$f:S^1\to X$$ such that $$f(0) = f(1) = x_0$$, modulo being homotopic maps.

• Loops can be homotopic (i.e. freely homotopic) without being homotopic rel a base point, so not equal in $$\pi_1(X; x_0)$$.

• Counterexample where homotopic loops are not equal in $$\pi_1$$, but just conjugate. Need nonabelian $$\pi_1$$ for conjugates to possibly not be equal, so take a torus:

\hfill


$$\implies$$:

• Suppose $$\gamma_1 \simeq\gamma_2$$, then there exists a free homotopy $$H: I\times S^1 \to X$$ with $$H_0 = \gamma_0, H_1 = \gamma_1$$.

• Since $$H(0, 1) \gamma_0(1) = x_0$$ and $$H(1, 1) = \gamma_1(1) = x_0$$, the map \begin{align*} T: [0, 1] &\to X \\ t &\mapsto H(t, 1) \end{align*} descends to a loop $$T:S^1\to X$$.

• Claim: $$\gamma_1$$ and $$T\ast \gamma_2 \ast T^{-1}$$ are homotopic rel $$x_0$$, making $$\gamma_1, \gamma_2$$ conjugate in $$\pi_1$$.

• Idea: for each fixed $$s$$, follow $$T$$ for the first third, $$\gamma_2$$ for the middle third, $$T^{-1}$$ for the last third.

$$\impliedby$$:

• Suppose $$[\gamma_1] = [h] [\gamma_2] [h]^{-1}$$ in $$\pi_1(X; x_0)$$. The claim is that $$\gamma_1 \simeq h\gamma_2 h^{-1}$$ are freely homotopic.
• Since these are equal in $$\pi_1$$, we get a square interpolating $$\gamma_1$$ and $$h\gamma_2 h^{-1}$$ with constant sides $$\operatorname{id}_{x_0}$$.
• For free homotopies, the sides don’t have to be constant, to merge $$h$$ and $$h^{-1}$$ into the sides to get a free homotopy from $$f$$ to $$g$$:

## 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

Suppose that $$X$$ is a topological space and $$x_0\in X$$, and suppose that every continuous map $$\gamma: S^1 \to X$$ is freely homotopic to the constant map to $$x_0$$. Prove that $$\pi_1(X, x_0) = \left\{{ e }\right\}$$.

Note that “freely” means there are no conditions on basepoints.

#topology/qual/completed #topology/qual/work #4