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Connected sum

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Any two surfaces can be attached to each other by punching holes in them and then gluing them together along these edges.

The result is called the connected sum $S_1 \# S_2$ of surfaces $S_1$, $S_2$.

For example, this is how you create the double torus by attaching two tori to each other in this fashion.

Double torus construction.jpg

More precisely, one considers these diagrams:

Double torus construction diagram.jpg

Then one can interpret the diagram by gluing along the edges:

These are the cuts:

Double torus construction cutting.jpg

Exercise. Prove $S \# {\bf S}^2 = S$.

Exercise. What is ${\bf P}$$^2 \# {\bf P}^2$?

Theorem (Classification of surfaces). (1) A compact connected surface is homeomorphic to

(2) These options are not homeomorphic.

The options are: ${\bf S}^2, n{\bf T}^2, n{\bf P}^2$.

Exercise. Classify:

  • ${\bf S}^2 \# {\bf S}^2,$
  • ${\bf P}^2 \# {\bf K}^2,$
  • ${\bf T}^2 \# {\bf K}^2,$
  • ${\bf K}^2 \# {\bf K}^2.$

This is a particular case of surgery: