Exercises:Mond - Topology - 2/Section B/Question 6

Section B

Question 6

$\xymatrix{ \bullet \ar@{<-}[dd]_B \ar@{<.}@/^1em/[rrr]^{(A)} \ar@{<-}[r]_{A_1} & \bullet \ar@{<-}[r]_{A_2} \ar@{}[ddr]|(.5){\mathbf{M} } \ar@{-->}@/_.65em/[dd]^{C_1} \ar@{-->}@/^.65em/[dd] & \bullet \ar@{<-}[r]_{A_3} \ar@{-->}@/_.65em/[dd]^{C_2} \ar@{-->}@/^.65em/[dd] & \bullet \\ & & &\\ \bullet \ar@{.>}@/_1em/[rrr]_{(A)} \ar[r]^{A_3} & \bullet \ar[r]^{A_2} & \bullet \ar[r]^{A_1} & \bullet \ar@{<-}[uu]_B }$ We take $\mathbb{RP}^2$ (represented here as the gluings $A$ and $B$) and cut it (creating gluings $C_1$ and $C_2$), thus diving $A$ into $A_1$, $A_2$ and $A_3$. The central strip is called $M$

$M$ is a Mobius strip inside $\mathbb{RP}^2$

The picture on the right shows that $\mathbb{RP}^2$ contains a Mobius strip, $M$. Use the diagram, taking into account the glueings, to describe the complement of $M$ in $\mathbb{RP}^2$. You mare allowed to cut it, provided you then glue it back together.

Complete the following sentence in as clear a way as possible:

• "$\mathbb{RP}^2$ is obtained from a Mobius strip by ....."