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

Section b

Question 6

[ilmath]\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 }[/ilmath] We take [ilmath]\mathbb{RP}^2[/ilmath] (represented here as the gluings [ilmath]A[/ilmath] and [ilmath]B[/ilmath]) and cut it (creating gluings [ilmath]C_1[/ilmath] and [ilmath]C_2[/ilmath]), thus diving [ilmath]A[/ilmath] into [ilmath]A_1[/ilmath], [ilmath]A_2[/ilmath] and [ilmath]A_3[/ilmath]. The central strip is called [ilmath]M[/ilmath]

[ilmath]M[/ilmath] is a Mobius strip inside [ilmath]\mathbb{RP}^2[/ilmath]

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

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

• "[ilmath]\mathbb{RP}^2[/ilmath] is obtained from a Mobius strip by ....."