Handle decompositions of surfaces

When a $1$-handle is attached to a single boundary circle of a handlebody, there are two standard ways of attaching it up to homeomorphism, pictured in Figure 2.17: on the left, the new boundary has two components in place of one; on the right, the new boundary still consists of one boundary circle, and there is a Möbius band embedded in the surface using a piece of the collar and the handle $h^{1}$.

Attaching a 1-handle to one boundary circle.

Find the Möbius band that is referred to above on the right surface in Figure 2.17.

Show that the result of attaching a 1-handle to the same boundary circle is never homeomorphic to what you get when it is attached to two different boundary circles.

Theorem: Let $H$ be a handlebody with a handle decomposition with $0$-handles and $1$-handles. Then: (a) $\partial H$ is homeomorphic to the disjoint union of a finite number of circles; (b) there is a neighborhood $N(\partial H)$ of $\partial H$ which is homeomorphic to $\partial H \times I$, where $\partial H \times \left\{ 0\right\} \subset \partial H \times I$ is sent to $\partial H \subset H$ via the standard identification $(x,0) \mapsto x$.

The proof outline is by induction on the number of $1$-handles: the case of $0$-handles alone is immediate, and for the inductive step one writes $H = H_{1} \cup h^{1}$, assumes (a) and (b) hold for $H_{1}$, and uses the existence of a collar where the $1$-handle is attached to determine the new boundary and its neighborhood, treating separately the case where the two attaching arcs of $h^1$ lie on the same boundary circle of $H_1$ and the case where they lie on two different boundary circles of $H_1$.

Fill in the details of this proof.

Show that a handlebody with only one $0$-handle is connected.

In the connected handlebodies of Figure 2.70, find a new decomposition with exactly one $0$-handle.

Decompositions with a single 0-handle.

Find a new handle decomposition for Figure 2.70(c) so that the $1$-handles are all attached to the boundaries of the $0$-handles.

What is the minimal number of $1$-handles required for a handle decomposition of $T \# T$? Construct a handle decomposition of $T \# T$ with this minimal number of $1$-handles.

Construct a handle decomposition of $(T \# T)_{(1)}$ with three $0$-handles, seven $1$-handles, and one $2$-handle.

Give a homeomorphism between $D^{1} \times D^{1} \cup_{f} h^{1}$ and $D^{1} \times D^{1} \cup_{g} h^{1}$, where $f(-1,x) = (-1,x)$, $f(1,x) = (1,x)$, and $g(-1,x) = (1,x)$, $g(1,x) = (-1,x)$.

Give a homeomorphism $h$ between $D^{1} \times D^{1} \cup_{f} h^{1}$ and $D^{1} \times D^{1} \cup_{g} h^{1}$, where $f(-1,x) = (-1,x)$, $f(1,x) = (1,x)$, $g(-1,x) = (x,1)$, $g(1,x) = (x,-1)$, and, moreover, $h$ restricts to the identity on a small disk in the interior of $D^{1} \times D^{1}$ as well as on $h^{1}$.