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.