Miscellaneous Algebraic Topology

1 (Fall ’14) #topology/qual/work

Prove that \({\mathbb{R}}^2\) is not homeomorphic to \({\mathbb{R}}^n\) 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 \(f : {\mathbb{RP}}^2 \to S^1 \times S^1\) is necessarily null-homotopic.

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

  • Identify \(\pi_1 {\mathbb{RP}}^2 = {\mathbf{Z}}/2{\mathbf{Z}}\) 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 \(\widetilde{S^1\times S^1} = {\mathbb{R}}^2\).

4 (Fall ’11) #topology/qual/completed

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

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

  • Identify \(\pi_1{\mathbb{RP}}^n = {\mathbf{Z}}/2{\mathbf{Z}}\) 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 \(S^2 \to {\mathbb{RP}}^2\) 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 \(f : S^3 \times S^3 \to {\mathbb{RP}}^3\) 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 \({\mathbb{R}}^3\) to itself such that \(f (\mathbf{x}) \perp \mathbf{x}\) (as vectors in \({\mathbb{R}}^3\) 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 \(X \subset {\mathbb{R}}^2\) which is not a retract of \({\mathbb{R}}^2\) .

  • 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