4
$\begingroup$

This is the second half of exercise 1.3.21 from Hatcher.

Let $Y$ be the space obtained by attaching a Möbius band $M$ to $\mathbb{R}P^2$ via a homeomorphism from its boundary circle to a circle in $\mathbb{R}P^2$ lifting to the equator in the covering space $S^2$ of $\mathbb{R}P^2$. Compute $\pi_1(Y)$, describe its universal cover, and describe the action of $\pi_1(Y)$ on the universal cover.

I've done the first half of the exercise, but this half has me a little mixed up. Mostly I'm just looking for a hint or a nudge in the right direction. Would the cell structure be something like: one $0$-cell, two $1$-cells, and two $2$-cells?

Thanks

  • 1
    I know it has been 5 years, but this question is probably relevant to other people too. This question is actually answered in https://math.stackexchange.com/questions/173023/universal-cover-of-projective-plane-glued-to-m%C3%B6bius-strip All you need to understand is that the circle lifting to the equator is the boundry of the 2-cell in $\Bbb R P^2$ that is seen as one 1-cell and one 2-cell attached via a map of degree 2.2017-10-15

0 Answers 0