Exercises:Mond - Topology - 2/Section B/Question 7

Section B

Question 7

Both the Klein bottle and the real projective plane are [ilmath]2[/ilmath]-manifolds. This is not obvious from their descriptions as quotients of the square by an equivalence relation. In fact each point [ilmath]x[/ilmath] does have an open neighbourhood homeomorphic to an open set in [ilmath]\mathbb{R}^2[/ilmath]. Show by carefully labelled drawings that this is true if:

1. [ilmath]x[/ilmath] is in the image of the interior of the square.
   • Caution:Mond VERY PROBABLY ALMOST CERTAINLY means the interior of the square considered as a set in [ilmath]\mathbb{R}^2[/ilmath], as of course the square's interior is itself when considered as a topological subspace
2. [ilmath]x[/ilmath] is in the image of an edge in the square, but not of a vertex.
3. [ilmath]x[/ilmath] is the image of a vertex.

Your drawings for 2 and 3 have to make use of the fact that the edges of the square are glue together in passing to the quotient.

Solution

Definitions

We take the following diagrams as the "square-edge-gluing" (identification?) diagrams used:

The Klein bottle	[ilmath]\mathbb{RP}^2[/ilmath]
[ilmath]\xymatrix{ \bullet \ar@{<-}[r]^A \ar[d]_B & \bullet \ar[d]^B \\ \bullet \ar@{->}[r]_A & \bullet }[/ilmath]	[ilmath]\xymatrix{ \bullet \ar@{<-}[r]^A \ar@{<-}[d]_B & \bullet \ar[d]^B \\ \bullet \ar@{->}[r]_A & \bullet }[/ilmath]