# Classification Theorem for Compact Surfaces

This is a Helper Page

This is a preliminary page that needs development.

Taken by Htasoff 17:00, 18 July 2011 (UTC). Rough diagram of how the connected sum of a Real Projective Plane and a Torus is homeomorphic to that of a Real Projective Plane and a Klein bottle. Since a Klein bottle is homeomorpic to the connected sum of two Projective Planes (not demonstrated), the connected sum of a Real Projective Plane and a Torus is homeomorphic to that of three Real Projective Planes.

Thus, as stated in the pictures (# means: the connected sum):

$RPP~\#~Torus = RPP~\#~Klein~bottle$

And (not described in the diagram):

$RPP~\#~RPP = Klein~bottle$

Thus:

$RPP~\#~Torus = RPP~\#~RPP~\#~RPP$

It is crucial to note, however, that you cannot simply subtract a RPP from both sides, because:

$Torus \neq Klein~bottle$

Thus, the connected sum of any number of tori and real projective planes can be reduced solely to a sum of real projective planes. This proof turns the and/ or to exclusively to or when added to the proof that any closed surface is homeomorphic to a sphere, and/ or to a connected sum of tori, and/ or to a connected sum of projective planes (also not yet proven on this page).

This diagram is based off of the book in this footnote[1].

I hope this will lead to an explanation of the claims made in the Why It's Interesting sections of the Real Projective Plane and Torus pages.

## Instructions for the Future

This page need to be written. I believe this to be interesting, and worth devoting a page to. Unfortunately, due to time constraints, I only had time to create the page. Here is what's needed:

• A quality visual proof like the one in the pictures or the book I've referenced.
• An explanation of how the location of the twist in the Mobius strip modeling the RPP renders the Klein bottle and torus effectively the same.
• I recommend using the 'equations' I wrote in this explanation.
• A proof of the broader statement: that any closed surface is homeomorphic to a sphere, and/ or to a connected sum of tori, and/ or to a connected sum of projective planes.
• Further development as you see fit.

## References

1. Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.