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 2-cells 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 2-sphere $$S^2$$ 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 2-sphere 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 two-sheeted 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|}(x-1)^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 2-cell 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 2-disk $${\mathbb{D}}^2$$ 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 2-cell 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 2-cell 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.

#topology/qual/work #7 #topology/qual/completed