Cutting and Pasting Example
Katie Gedeon
1. Show that the connected sum of a torus T and the projective plane RP2 is homeomorphic to the
connected sum of three copies of RP2 .
Proof. Let T #RP2 be given in the first picture below. Then by a chain of cuttings and gluings:
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We see that T #RP2 ' RP2 #RP2 #RP2 as desired.
2. Let X be a surface obtained by pasting edges of an 8-sided polygon with labeling scheme
−1
a1 a2 a3 a4 a1 a−1
4 a3 a2 .
To which standard surface is X homeomorphic?
Proof. We consider the polygon with the given labeling scheme, and follow a chain of cuttings and
gluings:
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Hence, X is homeomorphic to the connected sum of three copies of RP2 .
3. I’m not sure what problem I was working on this code for, but here it is anyway.
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