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.