# Miscellaneous Algebraic Topology

## 1 (Fall ’14) #topology/qual/work

Prove that $${\mathbf{R}}^2$$ is not homeomorphic to $${\mathbf{R}}^n$$ for $$n > 2$$.

## 2 (Spring ’12) #topology/qual/work

Prove that any finite tree is contractible, where a tree is a connected graph that contains no closed edge paths.

## 3 (Spring ’13) #topology/qual/completed

Show that any continuous map $$f : {\mathbf{RP}}^2 \to S^1 \times S^1$$ is necessarily null-homotopic.

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• Two techniques:
• Show $$f_* = 0$$
• Lift to a contractible universal cover.
• Any continuous map $${\mathbf{RP}}^2 \xrightarrow{f} S^1\times S^1$$ induces a group morphism $$\pi_1 {\mathbf{RP}}^2 \xrightarrow{f_*} \pi_1(S^1\times S^1)$$

• Identify $$\pi_1 {\mathbf{RP}}^2 = {\mathbf{Z}}/2{\mathbf{Z}}$$ and $$\pi_1(S^1\times S^1) = \pi_1 S^1 \times\pi_1 S^1 = {\mathbf{Z}}^2$$.

• But as a $${\mathbf{Z}}{\hbox{-}}$$module morphism, $$f_*$$ will preserve torsion submodules, and since $${\mathbf{Z}}^2$$ is free we must have $$f_* = 0$$.

• Lemma: $$f_* = 0$$ implies $$f$$ is nullhomotopic.

\todo[inline]{Why? What is the homotopy?}

• Note that $$\widetilde{S^1\times S^1} = {\mathbf{R}}^2$$.

## 4 (Fall ’11) #topology/qual/completed

Prove that, for $$n \geq 2$$, every continuous map $$f: {\mathbf{RP}}^n \to S^1$$ is null-homotopic.

\hfill

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• Any continuous map $${\mathbf{RP}}^n \xrightarrow{f} S^1$$ induces a group morphism $$\pi_1{\mathbf{RP}}^n \xrightarrow{f_*} \pi_1S^1$$

• Identify $$\pi_1{\mathbf{RP}}^n = {\mathbf{Z}}/2{\mathbf{Z}}$$ and $$\pi_1S^1 = {\mathbf{Z}}$$ to obtain a group morphism $$f_*: {\mathbf{Z}}/2{\mathbf{Z}}\to {\mathbf{Z}}$$.

• Claim: $$f_* = 0$$.

• Recognizing this as a map of $${\mathbf{Z}}{\hbox{-}}$$modules, we must have \begin{align*} 0 = [2]_2 = 2\cdot [1]_2 \implies 0 = f_*(0) = 2\cdot f_*([1]_2) .\end{align*} since $${\mathbf{Z}}{\hbox{-}}$$module maps send 0 to 0.

• But no element of the image $${\mathbf{Z}}$$ is annihilated by $$2$$, so $$f_*$$ can only be the zero map.

• But then $$f$$ is nullhomotopic.

• Lemma: $$f_* = 0$$ implies $$f$$ is nullhomotopic.

\todo[inline]{Why?}

## 5 (Spring ’06) #topology/qual/work

Let $$S^2 \to {\mathbf{RP}}^2$$ be the universal covering map.

Is this map null-homotopic? Give a proof of your answer.

## 6 (Spring ’17) #topology/qual/completed

Suppose that a map $$f : S^3 \times S^3 \to {\mathbf{RP}}^3$$ is not surjective.

Prove that $$f$$ is homotopic to a constant function.

\todo[inline]{Lost, redo.}


## 7 (Fall ’06) #topology/qual/work

Prove that there does not exist a continuous map $$f : S^2 \to S^2$$ from the unit sphere in $${\mathbf{R}}^3$$ to itself such that $$f (\mathbf{x}) \perp \mathbf{x}$$ (as vectors in $${\mathbf{R}}^3$$ for all $$\mathbf{x} \in S^2$$).

## 8 (Spring ’08) #topology/qual/work

Let $$f$$ be the map of $$S^1 \times [0, 1]$$ to itself defined by \begin{align*} f (e^{i\theta} , s) = (e^{i(\theta+2\pi s)} , s) ,\end{align*} so that $$f$$ restricts to the identity on the two boundary circles of $$S^1 \times [0, 1]$$.

Show that $$f$$ is homotopic to the identity by a homotopy $$f_t$$ that is stationary on one of the boundary circles, but not by any homotopy that is stationary on both boundary circles.

Hint: Consider what $$f$$ does to the path $$s \mapsto (e^{i\theta_0} , s)$$ for fixed $$e^{i\theta_0} \in S^1$$.

## 9 (Spring ’17) #topology/qual/work

Show that $$S^1 \times S^1$$ is not the union of two disks (where there is no assumption that the disks intersect along their boundaries).

## 10 (Spring ’14) #topology/qual/work

Suppose that $$X \subset Y$$ and $$X$$ is a deformation retract of $$Y$$.

Show that if $$X$$ is a path connected space, then $$Y$$ is path connected.

## 11 (Spring ’05) #topology/qual/work

Do one of the following:

• Give (with justification) a contractible subset $$X \subset {\mathbf{R}}^2$$ which is not a retract of $${\mathbf{R}}^2$$ .

• Give (with justification) two topological spaces that have the same homology groups but that are not homotopy equivalent.

## 12 (Spring ’16) #topology/qual/work

Recall that the suspension of a topological space, denoted $$SX$$, is the quotient space formed from $$X \times [-1, 1]$$ by identifying $$(x, 1)$$ with $$(y, 1)$$ for all $$x, y \in X$$, and also identifying $$(x, -1)$$ with $$(y, -1)$$ for all $$x, y \in X$$.

• Show that $$SX$$ is the union of two contractible subspaces.

• Prove that if $$X$$ is path-connected then $$\pi_1 (SX) = \left\{{0}\right\}$$.

• For all $$n \geq 1$$, prove that $$H_{n} (X) \cong H_{n+1} (SX)$$.
#topology/qual/work #topology/qual/completed