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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.

More precisely, one considers these diagrams:

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

These are the cuts:

**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

- the sphere ${\bf S}^2$, or
- the connected sum of $n$ tori ${\bf T}^2$, or
- the connected sum of $n$ projective planes ${\bf P}^2$.

(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*: