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Sketch a handlebody decomposition on T2#T2 with three 0-handles, six 1-handles, and one 2-handle hints:...

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Accounting

Sketch a handlebody decomposition on T2#T2 with three 0-handles, six
1-handles, and one 2-handle
hints:
If you wish to maximize the fun, try not to read this hint ! But whatever you
do, explain why you are left with a disk when you take away the 0-handles and
1-handles that you have sketched on T2#T2
.
Find two subsets A and B of X = T2#T2 each of which is topologically
equivalent to an annulus (that is, a band) and have the property that
X \ (A B) is topologically equivalent to S2 with four holes. (You may
want to illustrate this with a sequence of sketches.) Notice that each annulus
is a 0-handle with a 1-handle attached. Place a disk D (another 0-handle)
on X which is disjoint from A and B. Then X \(AB D) is topologically
equivalent to S2 with five holes. Now run four more 1-handles through this
sphere with five holes so that the remainder consists of one single disk, which
is your 2-handle. Draw all your handles into a depiction of X, including
the 2-handle.

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