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\large{2014 Fall Math 5041 \quad Assignment 1. \qquad Due: Sept 26, 2014}
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\large{Name: }
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%1. Show explicitly that $\mathbb{R}P^3$ is null-cobordant.
In problems 1 and 2, we will show explicitly that $\mathbb{R}P^3$ is null-cobordant.
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1.
a) Consider the set of quanternions
$$
\mathbb{H} = \{x = a + bi + cj + dk ~|~ a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j \}.
$$
It is clear that $\mathbb{H}$ is a real vector space. We define the conjugate of $x = a + bi + cj + dk$ to be $\overline{x} = a - bi - cj - dk$. Show that $x \mapsto ||x|| = \sqrt{x\overline{x}}$ defines a norm on $\mathbb{H}$.
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b) Show that $\mathbb{H} \setminus \{0\}$ is a group, and the unit sphere
$$
U = \{x \in \mathbb{H} ~|~ ||x|| = 1\}
$$
is a subgroup.
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Remark: $U$ is (isomorphic to) the so-called $SU(2, \mathbb{C})$ group.
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c) Show that the action of $U$ on $\mathbb{H}$ by conjugation
\begin{equation} \label{eq: conj}
g \cdot x \mapsto g x g^{-1}
\end{equation}
preserves the norm, and
$$
V = \{a + bi + cj + dk \in \mathbb{H} ~|~ a = 0\}
$$
is an invariant subspace.
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d) Show that the stabilizer of the action of $U$ on $V$ is $\{\pm 1\}$. Use this result to conclude that the special orthogonal group
$$
SO(3, \mathbb{R}) = \{A \in \mathtt{Mat}_{3\times 3}(\mathbb{R}) ~|~ A A^T = I \}
$$
is diffeomorphic to $\mathbb{R}P^3$.
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Remark: This is the well-known fact that $SU(2)$ is the universal double cover of $SO(3)$.
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2. Consider the subset of $T\mathbb{R}^3$ consisting of the vectors tangent to the 2-sphere $S^2 \subset \mathbb{R}^3$ of unit length. Here, we use the usual Euclidean length on $\mathbb{R}^3$, and the fact that
$$
T\mathbb{R}^3 \cong \mathbb{R}^3 \times \mathbb{R}^3.
$$
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a) Prove this subset $M$ is a submanifold.
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b) Show that $M$ is diffeomorphic to $SO(3, \mathbb{R})$, and therefore diffeomorphic to $\mathbb{R}P^3$.
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c) Show that $\mathbb{R}P^3$ is null-cobordant.
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3. Consider $H^1 = [0, \infty) \subset \mathbb{R}$ with the induced smooth structure. Clearly, $H^1$ is a manifold with boundary. Now consider the smooth map
$$
f: (0, 1) \to (0, 1), \qquad x \mapsto x^2.
$$
Define
$$
\widetilde{H^1} = \frac{U \sqcup V}{\sim}
$$
where $U = [0, 1)$, $V = (0, \infty)$, and for $x \in (0, 1) \subset V$, $y \in (0, 1) \in U$, $x \sim y$ if $y = f(x)$.
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a) Show that $\widetilde{H^1}$ is a smooth manifold with boundary.
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b) Show that the map
$$
V \to H^1, x \to x
$$
extends to a homeomorphism from $\widetilde{H^1}$ to $H^1$ which is not a diffeomorphism.
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4. The Hopf fibration is defined as follows. Let $S^3$ be embedded as the unit sphere in
$$
\mathbb{R}^4 \setminus \{0\} \cong \mathbb{C}^2 \setminus \{0\}
$$
Recall we have the canonical projection
$$
\pi: \mathbb{C}^2 \setminus \{0\} \to \mathbb{C}P^1
$$
and the isomorphism $\mathbb{C}P^1 \cong S^2$. Therefore, we obtain an induced projection
\begin{equation} \label{eq: hopf}
\pi: S^3 \to S^2
\end{equation}
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a) Express \eqref{eq: hopf} explicitly in coordinates. (One chart on each of $S^3$ and $S^2$ is enough!)
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b) Show that this is a fiber bundle with fiber $S^1$. (This is called the Hopf fibration.)
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5. Show that the tangent bundle of the 2-torus $T(T^2)$ is diffeomorphic to $T^2 \times \mathbb{R}^2$.
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Remark: Not all tangent bundles are trivial, e.g. $T(S^2)$ is not diffeomorphic to $S^2 \times \mathbb{R}^2$. In general, if the tangent bundle of $M$ is trivial, then we say $M$ is parallelizable.
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