(a) If \(M\) is a manifold-with-boundary, the tangent bundle \(T M\) is defined
exactly as for \(M\); elements of \(M_{p}\) are \(\tilde{p}\) equivalence classes of
pairs \((x, v)\). Although \(x\) takes a neighborhood of \(p \in \partial M\) onto
\(\mathbb{H}^{n}\), rather than \(\mathbb{R}^{n}\), the vectors \(v\) still run
through \(\mathbb{R}^{n}\), so \(M_{p}\) still has tangent vectors "pointing in
all directions". If \(p \in \partial M\) and \(x: U \rightarrow \mathbb{H}^{n}\)
is a coordinate system around \(p\), then
\(x_{*}^{-1}\left(\mathbb{R}^{n-1} x(p)\right) \subset M_{p}\) is a subspace.
Show that this subspace does not depend on the choice of \(x\); in fact, it is
\(I_{*}(\partial M)_{p}\), where \(i: \partial M \rightarrow M\) is the inclusion.
(b) Let \(a \in \mathbb{R}^{n-1} \times\\{0\\} \subset \mathbb{M}^{n} .\) A
tangent vector in \(\mathbb{M}^{n}{ }_{a}\) is said to point "invard" if, under
the identification of \(T \mathbb{M}^{n}\) with
\(\varepsilon^{n}\left(\mathbb{M}^{n}\right)\), the vector is \((a, v)\) where
\(v^{n}>0 .\) A vector \(v \in M_{p}\) which is not in \(i_{*}(\partial M)_{p}\) is
said to point "inward" if
\(x_{*}(v) \in \mathbb{M}^{n} x(p)\) points inward. Show that this definition
does not depend on the coordinate system \(x\).
(c) Show that if \(M\) has an orientation \(\mu\), then \(\partial M\) has a unique
orientation \(\partial \mu\) such that \(\left[v_{1}, \ldots,
v_{n-1}\right]=(\partial \mu)_{p}\) if and only if \(\left[w, i_{*} v_{1},
\ldots, i_{*} v_{n-1}\right]=\mu_{p}\) for every outward pointing \(w \in M_{p}\)
(d) If \(\mu\) is the usual orientation of \(\mathbb{H}^{n}\), show that \(\partial
\mu\) is \((-1)^{n}\) times the usual orientation of \(\mathbb{R}^{n-1}=\partial
\mathbb{H}^{n} .\) (The reason for this choice will become clear in Chapter 8.)
(e) Suppose we are in the setup of Problem 2-14. Define \(g: \partial M
\times[0,1) \rightarrow\) \(\partial N \times[0,1)\) by \(g(p, t)=(f(p), t) .\)
Show that \(T P\) is obtained from \(T M \cup T N\)
$$
v \in(\partial M)_{p} \quad \text { with }\left(\beta^{-1}\right)_{*} g_{*}
\alpha_{*}(v) \in(\partial N)_{f(p)}
$$
(f) If \(M\) and \(N\) have orientations \(\mu\) and \(\nu\) and \(f:(\partial M,
\partial \mu) \rightarrow(\partial N, \partial \nu)\) is orientation-reversing,
show that \(P\) has an orientation which agrees with \(\mu\) and \(\nu\) on \(M
\subset P\) and \(N \subset P\).
(g) Suppose \(M\) is \(S^{2}\) with two holes cut out, and \(N\) is \([0,1] \times
S^{1}\). Let \(f\) be a diffeomorphism from \(M\) to \(N\) which is orientation
preserving on one copy of \(S^{1}\) alnd orientation reversing on the other.
What is the resulting manifold \(P ?\)
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