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Solution to Proble

LetK= Xaba1b be the Klein bottle.(a)Find a representation for the fundamental group (b)Find a double covering map p :TK, whereTphism of the fundamental groups.

(a) Begin by labeling one of the vertices, say the bottvertices are identified with x0:

a

a

b b

x0 a

a

b

x0

x0

Since the edges labeled a are identified, the bottom both are the beginning of where edges labeled a be

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We now compute the homomorphism p: 1(T, x0)

generated by the loops =a1a2 and = b1. From the

pa1= a, pa2=a, pb1= b

The relation aba1b =1 in 1(K, y0) insures that theon 1(T, x0), i.e.

(p)(p) = (p