Classification theorem for closed surfaces

A closed surface is a compact, connected metrizable 2-manifold without boundary.

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:

Identify the surface depicted in Figure 2.57.

Surface for Exercise 2.7.8.

In this and the following problems, you are asked to identify surfaces; that is, tell which one of the standard model surfaces $S_{(p)}$, $T^{(g)}_{(p)}$, $P^{(h)}_{(p)}$ the given surface is homeomorphic to.

Which surface is the surface of a coffee cup (with a handle) homeomorphic to?

Identify the following surface. Take an empty box (with thickness) and poke a small hole in each of its six faces. The surface is the surface of the box after the holes are poked.

Identify the surfaces in Figure 2.69 (see Exercise 2.9.41).

Finding a Möbius band.

Identify the surfaces in Figure 2.70 (see Exercise 2.9.43).

Decompositions with a single 0-handle.

Identify the surfaces in Figure 2.71 (see Exercise 2.9.47).

Connected sums.

Identify the surfaces in Figure 2.72 (see Exercise 2.9.48).

Separating arcs and boundary sums.

Identify the surfaces in Figure 2.73 (see Exercise 2.9.49).

Separating circles and connected sums.

Identify the surface in Figure 2.78.

Surface for Exercise 2.9.70.

Identify the surface in Figure 2.79.

Surface for Exercise 2.9.71.

Identify the surface in Figure 2.80.

Surface for Exercise 2.9.72.

Show that the genus of $P^{(k)}$ is $k$.

Suppose $F$ is a functor from topological spaces and continuous maps to finitely generated abelian groups and homomorphisms so that for compact connected surfaces (without boundary) $A, B$, $F(A \# B) = F(A) \oplus F(B)$. For finitely generated abelian groups, it is true that $G_1 \oplus G_2$ is isomorphic to $G_1$ iff $G_2 = \left\{ e\right\}$. Show that this implies that $F(S)$ is the trivial group, where $S$ denotes the sphere. Show that if $F(T)$ is not $\left\{ e\right\}$, where $T$ denotes the torus, then $F$ distinguishes $T^{(g)}$, $g \geq 0$; that is, $F(T^{(p)})$ is not isomorphic to $F(T^{(q)})$, for $p \neq q$, $p, q \geq 0$.

Compute $\pi_1^{ab}(M)$ for the surfaces in the figure below, and use your results and the number of boundary components, to classify the surfaces.

Surfaces for Exercise 3.9.7.