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What is the surface of T#P#P? Where T=S^1xS^1(torus), P be a polygonal region in the plane....

What is the surface of T#P#P?

Where T=S^1xS^1(torus), P be a polygonal region in the plane.

question about algebra topology.

Math   Advanced-Math   
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Answer #1

, a polygonal region in the plane.

Now the surface is a two polygonal regions attached to a torus. The attachment is done by removing an open disk from each surface and attaching them using a homeomorphism of the disk boundary.

A polygonal region in the plane when added to any surface creates a hole in the surface. This is because any such region is homeomorphic to a disk and we are cutting a hole in this disk and attaching to a surface.

Hence the final surface is a torus with two closed holes cut out.

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