Algebraic topology question

• Dec 23rd 2010, 07:55 PM
Linnus
Algebraic topology question
Hey, I need some help with part 2 of this question.

i) compute $\displaystyle \Pi_1(\Re P^2)$
I got $\displaystyle Z_2$ for this one.

here is part 2.
ii) Describe a map $\displaystyle S^1 \rightarrow \Re P^2$ with i* not equal to 0. Assume $\displaystyle \Pi_1 (S^1)=Z$ and $\displaystyle A \rightarrow X$ is a retraction means $\displaystyle r \cdot i = I_{da}$.

The i is suppose to be on top of the arrow, but I'm not sure how to do that with latex.

Thanks!
• Dec 24th 2010, 02:46 AM
xxp9
if I understood your question correctly, you need to find a non-trivial loop in RP^2. Consider a curve c connecting the north pole pn=(0,0,1) and the south pole ps=(0,0,-1) on a sphere S^2, and let p(c) be the image of this curve by the projection p: S^2 -> RP^2 which indetifies all antipodal points. Then p(c) is a closed loop since p(pn)=p(ps). If p(c) is trivial, there is a homotopy, that is, a family of closed curves c_t with the common base point p0=p(pn), with c_0=p0 the constant map and c_1 = p(c). Lift this family of curves we get a homotopy on S^2 that continously deform the c to a pole, while keeping the ending points of c fixed. This is a contradiction.