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.