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I'm trying to calculate the fundamental group of two Möbius strips which have been identified along their boundary (which is a Klein bottle, I think). I've chosen an NDR pair $A,B$ where $A$ and $B$ are each of the original Möbius strips. The intersection $A \cap B$ is the boundary of the Klein bottle. I've heard that this boundary, and each of the Möbius strips, are contractible to $S^1$, but I'm having trouble visualizing this. Any help would be appreciated. Thanks

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    The Klein bottle does not have a boundary.2012-05-22
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    The Möbius strip deformation retracts to a circle drawn down the middle. Imagine pushing down uniformly toward the central circle all along the strip.2012-05-22
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    @JimConant Oops, sorry. I mean to ask why does $A \cap B$ deformation retract onto $S^1$?2012-05-22
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    Aha, I assume you mean, $A$, $B$ are slightly larger versions of the original Möbius strips. They intersect in something which is homeomorphic to a strip running along the boundary of a Möbius strip. This deformation-retracts to the boundary circle of the Möbius strip by pushing towards the boundary.2012-05-22
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    @JimConant Is there a way to visualise these on the unit square representations?2012-05-22

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