The classification theorem

Let $B$ be the Möbius band. Construct a model of $B \# T$ (the connected sum of $B$ with the torus $T$) and show that it is homeomorphic to $B \# K$ (the connected sum of $B$ with the Klein bottle $K$).

Here $P^{(k)}_{(1)}$ denotes the boundary connected sum of $k$ copies of the projective plane $P$ with one disk removed, and similarly $T^{(k)}_{(1)}$ for $k$ copies of the torus $T$. Show that $P^{(k)}_{(1)} \mathbin {\overline{\cup }} T_{(1)} \simeq P^{(2k+1)}_{(1)}$. Conclude from this that $P^{(l)}_{(1)} \mathbin {\overline{\cup }} T^{(k)}_{(1)} \simeq P^{(2k+l)}_{(1)}$.

For each of the orientable surfaces (given by handle decompositions with a single $0$-handle and disjoint $1$-handles attached to its boundary) in Figure 2.46, use handle sliding to put it in standard form and identify the surface. Check your answer using the classification theorem (Theorem 2.6.2: a connected, orientable handlebody with $0$- and $1$-handles and Euler characteristic $\chi$ and $p$ boundary circles is homeomorphic to the standard model for $S_{(p)}$ or $T^{(g)}_{(p)}$, where $\chi = 2 - 2g - p$).

Surfaces for Exercise 2.6.3.

For each of the nonorientable surfaces in Figure 2.47, use handle sliding to put it in standard form and identify the surface. Check your answer using the classification theorem (Theorem 2.6.2: a connected, nonorientable handlebody with $0$- and $1$-handles and Euler characteristic $\chi$ and $p$ boundary circles is homeomorphic to the standard model for $P^{(h)}_{(p)}$, where $\chi = 2 - h - p$).

Surfaces for Exercise 2.6.4.

Consider the step in the proof of the classification theorem (Theorem 2.6.2) for the orientable case where a new torus pair (a pair of linked $1$-handles) is created by adding a new $1$-handle to a handlebody $H_{1}$ already in standard form $T^{(g)}_{(p)}$, giving $H = H_{1} \cup h^{1}$. Show that, if we regard the $0$-handle together with this new torus pair as giving us the torus-with-one-hole $T_{(1)}$, then we can remove this piece from the surface and replace it by a consistently oriented disk with the same boundary, to which the other handles (those of $H_{1}$ besides its $0$-handle) are attached. Considering this disk as the new $0$-handle and the other $1$-handles as attached to its boundary, forming a handlebody $H_{1}'$, show that $H \simeq H_{1}' \# T$. Then use induction to show that $H \simeq T^{(g+1)}_{(p)}$ when $H_{1}' \simeq T^{(g)}_{(p)}$.

Consider the step in the proof of the classification theorem (Theorem 2.6.2) for the orientable case where a new $1$-handle is attached to a single boundary circle of a handlebody $H_{1}$ (already in standard form) to form $H = H_{1} \cup h^{1}$. Show that if $H_{1}$ is homeomorphic to $T^{(g)}_{(p)}$, then we can add a $2$-handle to $H$ so that $H \cup h^{2} \simeq H_{1}$. Thus, removing this $2$-handle shows that $H \simeq T^{(g)}_{(p+1)}$.

Consider the step in the proof of the classification theorem (Theorem 2.6.2) for the nonorientable case where we attach a new $1$-handle $h^{1}$ to form $H$ from a handlebody $H_{1}$, which, by induction, may be assumed to be either $T^{(g)}_{(p)}$ or $P^{(h)}_{(p)}$ and already in standard form. Show that we can assume in either case that $h^1$ is attached away from the torus pairs or twisted handles of $H_1$, so it may be regarded as attached to $S_{(p)}$ (the disk with $p-1$ trivial handles). In the case where $H_{1}$ is orientable, $h^{1}$ will necessarily be attached to $S_{(p)}$ so that $S_{(p)} \cup h^{1}$ is nonorientable. Show that this is not necessarily the case when $H_{1}$ is nonorientable.

Show that if a $1$-handle $h^{1}$ is attached to a single boundary circle of $H_{1} \simeq S_{(p)}$ as a twisted handle to form $H$, then the union of the $0$-handle and $h^{1}$ forms a Möbius band $P_{(1)}$. Form a new handlebody $H_{2}$ by replacing $h^{0} \cup h^{1}$ by a disk with the same boundary and then attaching the other $1$-handles (those of $H_1$ besides its $0$-handle) to it. Show that $H_{2} \simeq S_{(p)}$ and $H \simeq P \# H_{2} \simeq P_{(p)}$.

Show that if a $1$-handle $h^{1}_{1}$ is attached to two boundary circles of $S_{(p)}$ in a twisted fashion to form $H$, and we choose another $1$-handle $h^{1}_{2}$ so that $h^{0} \cup h^{1}_{2}$ forms an annulus whose boundary consists of these two circles, then $H_{1} = h^{0} \cup h^{1}_{2} \cup h^{1}_{1}$ is a Klein bottle with a disk removed, $K_{(1)}$. Form a new handlebody $H_{2}$ by removing $h^{0} \cup h^{1}_{2} \cup h^{1}_{1}$ and replacing it with a disk (the new $0$-handle), then adding in all of the other handles of the original standard handle decomposition of $S_{(p)}$. Show that $H_{2} \simeq S_{(p-1)}$ and $H \simeq K \# S_{(p-1)}$.

Deduce the general induction step for the nonorientable case of the classification theorem (Theorem 2.6.2) from the constructions of the last three exercises (2.6.7--2.6.9), considering separately: