Cell Complexes and Adjunction Spaces

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 \({\mathbb{RP}}^2\) via a homeomorphism of its boundary circle to the standard \({\mathbb{RP}}^1\) in \({\mathbb{RP}}^2\).

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 \({\mathbb{R}}^3\) 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 \(S^2 = \left\{{\mathbf{x} \in {\mathbb{R}}^3 {~\mathrel{\Big\vert}~}{\left\lVert {\mathbf{x}} \right\rVert} = 1}\right\}\) 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 \(X = S_1 \cup S_2 \subset {\mathbb{R}}^3\) 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 \({\mathbb{RP}}^2\) via some map \(f: S^1 \to {\mathbb{RP}}^2\) .

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 \(\pi :{\mathbb{RP}}^N \to X\) 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