AlgTop21: The two-holed torus and 3-crosscaps surface

We describe how the two-holed torus and the 3-crosscaps surface can be given hyperbolic geometric structure. For the two-holed torus we cut it into 4 hexagons and then describe a tesselation of the hyperbolic plane (using the Beltrami Poincare model described in the previous lecture) composed of regular hexagons meeting four at a vertex. We will look at an octagon model involving the standard form. Then we briefly look at the 3-crosscaps surface in the same way.

This is the 21st lecture in this beginner's course on Algebraic Topology, given by N J Wildberger of UNSW.

Category:

Education

Tags:

• AlgTop21
• mathematics
• education
• algebraic
• topology
• course
• Wildberger
• two
• holed
• torus
• three
• crosscaps
• surface
• classification
• Euler
• characteristic
• hyperbolic
• geometry
• symmetries
• UNSW

[TheMathLife]

You can take connected sums of Tori and Projective planes too can't you?

[njwildberger]

@TheMathLife Yes you can.

[TheMathLife]

@njwildberger Ok, because your theorem at 7:05 makes it look like you can only add the Tori and Planes, but not mix them. I love the series so far by the way. I loved Point Set topology but found Algebraic Topology very difficult, your lectures are helping me to appreciate the subject more!