1 (Fall ’07) #topology/qual/work
Describe a cell complex structure on the torus \(T = S^1 \times S^1\) and use this to compute the homology groups of \(T\).
To justify your answer you will need to consider the attaching maps in detail.
2 (Fall ’04) #topology/qual/work
Let \(X\) be the space formed by identifying the boundary of a Möbius band with a meridian of the torus \(T^2\).
Compute \(\pi_1 (X)\) and \(H_* (X)\).
3 (Spring ’06) #topology/qual/work
Compute the homology of the space \(X\) obtained by attaching a Möbius band to \({\mathbf{RP}}^2\) via a homeomorphism of its boundary circle to the standard \({\mathbf{RP}}^1\) in \({\mathbf{RP}}^2\).
4 (Spring ’14) #topology/qual/work
Let \(X\) be a space obtained by attaching two 2cells to the torus \(S^1 \times S^1\), one along a simple closed curve \(\left\{{x}\right\} \times S^1\) and the other along \(\left\{{y}\right\} \times S^1\) for two points \(x \neq y\) in \(S^1\) .
a
Draw an embedding of \(X\) in \({\mathbf{R}}^3\) and calculate its fundamental group.
b
Calculate the homology groups of \(X\).
5 (Fall ’07) #topology/qual/work
Let \(X\) be the space obtained as the quotient of a disjoint union of a 2sphere \(S^2\) and a torus \(T = S^1 \times S^1\) by identifying the equator in \(S^2\) with a circle \(S^1 \times \left\{{p}\right\}\) in \(T\).
Compute the homology groups of \(X\).
6 (Spring ’06) #topology/qual/work
Let \(X = S^2 / \left\{{p_1 = \cdots = p_k }\right\}\) be the topological space obtained from the 2sphere by identifying \(k\) distinct points on it (\(k \geq 2\)).
Find:

The fundamental group of \(X\).

The Euler characteristic of \(X\).
 The homology groups of \(X\).
7 (Fall ’16) #topology/qual/work
Let \(X\) be the topological space obtained as the quotient of the sphere \(S^2 = \left\{{\mathbf{x} \in {\mathbf{R}}^3 {~\mathrel{\Big\vert}~}{\left\lVert {\mathbf{x}} \right\rVert} = 1}\right\}\) under the equivalence relation \(\mathbf{x} \sim \mathbf{x}\) for \(\mathbf{x}\) in the equatorial circle, i.e. for \(\mathbf{x} = (x_1, x_2, 0)\).
Calculate \(H_* (X; {\mathbf{Z}})\) from a CW complex description of \(X\).
8 (Fall ’17) #topology/qual/work
Compute, by any means available, the fundamental group and all the homology groups of the space obtained by gluing one copy \(A\) of \(S^2\) to another copy \(B\) of \(S^2\) via a twosheeted covering space map from the equator of \(A\) onto the equator of \(B\).
9 (Spring ’14) #topology/qual/work
Use cellular homology to calculate the homology groups of \(S^n \times S^m\).
10 (Fall ’09/Fall ’12) #topology/qual/work
Denote the points of \(S^1 \times I\) by \((z, t)\) where \(z\) is a unit complex number and \(0 \leq t \leq 1\). Let \(X\) denote the quotient of \(S^1 \times I\) given by identifying \((z, 1)\) and \((z_2 , 0)\) for all \(z \in S^1\).
Give a cell structure, with attaching maps, for \(X\), and use it to compute \(\pi_1 (X, \ast)\) and \(H_1 (X)\).
11 (Spring ’15) #topology/qual/work
Let \(X = S_1 \cup S_2 \subset {\mathbf{R}}^3\) be the union of two spheres of radius 2, one about \((1, 0, 0)\) and the other about \((1, 0, 0)\), i.e. \begin{align*} S_1 &= \left\{{(x, y,z) \mathrel{\Big}(x1)^2 + y^2 +z^2 = 4}\right\} \\ S_2 &= \left\{{(x, y, z) \mathrel{\Big}(x + 1)^2 + y^2 + z^2 = 4}\right\} .\end{align*}

Give a description of \(X\) as a CW complex.

Write out the cellular chain complex of \(X\).
 Calculate \(H_* (X; Z)\).
12 (Spring ’06) #topology/qual/work
Let \(M\) and \(N\) be finite CW complexes.

Describe a cellular structure of \(M \times N\) in terms of the cellular structures of \(M\) and \(N\).

Show that the Euler characteristic of \(M \times N\) is the product of the Euler characteristics of \(M\) and \(N\).
13 (Spring ’07) #topology/qual/work
Suppose the space \(X\) is obtained by attaching a 2cell to the torus \(S^1 \times S^1\).
In other words, \(X\) is the quotient space of the disjoint union of the closed disc \({\mathbb{D}}^2\) and the torus \(S^1 \times S^1\) by the identification \(x \sim f(x)\) where \(S^1\) is the boundary of the unit disc and \(f : S^1 \to S^1 \times S^1\) is a continuous map.
What are the possible homology groups of \(X\)? Justify your answer.
14 (Spring ’15) #topology/qual/work
Let \(X\) be the topological space constructed by attaching a closed 2disk \({\mathbb{D}}^2\) to the circle \(S^1\) by a continuous map \(\partial{\mathbb{D}}^2 \to S^1\) of degree \(d > 0\) on the boundary circle.

Show that every continuous map \(X \to X\) has a fixed point.

Explain how to obtain all the connected covering spaces of \(X\).
15 (Spring ’11) #topology/qual/work
Let \(X\) be a topological space obtained by attaching a 2cell to \({\mathbf{RP}}^2\) via some map \(f: S^1 \to {\mathbf{RP}}^2\) .
What are the possibilities for the homology \(H_* (X; Z)\)?
16 (Spring ’12) #topology/qual/work
For any integer \(n \geq 2\) let \(X_n\) denote the space formed by attaching a 2cell to the circle \(S^1\) via the attaching map \begin{align*} a_n: S^1 &\to S^1 \\ e^{i\theta} &\mapsto e^{in\theta} .\end{align*}
a
Compute the fundamental group and the homology of \(X_n\).
b
Exactly one of the \(X_n\) (for \(n \geq 2\)) is homeomorphic to a surface. Identify, with proof, both this value of \(n\) and the surface that \(X_n\) is homeomorphic to (including a description of the homeomorphism).
17 (Spring ’09) #topology/qual/work
Let \(X\) be a CW complex and let \(\pi : Y \to X\) be a covering space.
a
Show that \(Y\) is compact iff \(X\) is compact and \(\pi\) has finite degree.
b
Assume that \(\pi\) has finite degree \(d\). Show show that \(\chi (Y ) = d \chi (X)\).
c
Let \(\pi :{\mathbf{RP}}^N \to X\) be a covering map. Show that if \(N\) is even, \(\pi\) is a homeomorphism.
18 (Spring ’18) #topology/qual/work
For topological spaces \(X, Y\) the mapping cone \(C(f )\) of a map \(f : X \to Y\) is defined to be the quotient space \begin{align*} (X \times [0, 1]){\textstyle\coprod}Y / \sim &{\quad \operatorname{where} \quad} \\ (x, 0) &\sim (x', 0) {\quad \operatorname{for all} \quad} x, x' \in X \text{ and } \\ (x, 1) &\sim f (x) {\quad \operatorname{for all } \quad} x \in X .\end{align*}
Let \(\phi_k : S^n \to S^n\) be a degree \(k\) map for some integer \(k\).
Find \(H_i(C(\phi_k ))\) for all \(i\).
Spring 2019 #7 #topology/qual/completed
For \(f:X\to Y\), the mapping cone of \(f\) is defined as \begin{align*} C_f \coloneqq\qty{X\times I} {\textstyle\coprod}Y/\sim \\ (x, 0) \sim (x', 0) \quad \text{for all }x, x'\in X\\ (x, 1) \sim f(x) .\end{align*}
Let \(\phi_k: S^1\to S^1\) be a \(k{\hbox{}}\)fold covering and find \(\pi_1\qty{C_f}\).
\todo[inline]{Revisit, old. Maybe redo.}
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Let \(f: S^1 \xrightarrow{\times k} S^1\).
Claim: The inclusion \(S^1 \to C_\phi\) induces an isomorphism \(\pi_1(C_\phi) \cong \pi_1(S^1) / H\) where \(H = N_{\pi_1(S^1)}(\left\langle{f^*}\right\rangle)\) is the normal subgroup generated by the induced map \(f^* \pi_1(S^1) \to \pi_1(S^1)\).

Since \(f\) is a \(k{\hbox{}}\)fold cover, the induced map is multiplication by \(k\) on the generator \(\alpha \in \pi_1(S^1)\), i.e. \(\alpha \mapsto \alpha^k\).

But then \(\pi_1(S^1) \cong {\mathbf{Z}}\) and \(H \cong k{\mathbf{Z}}\), so \(\pi_1(C_\phi) \cong {\mathbf{Z}}/m{\mathbf{Z}}\).
19 (Fall ’18) #topology/qual/work
Prove that a finite CW complex must be Hausdorff.