# Relations between the real projective space and canonical covering projections

• Mar 17th 2012, 02:41 PM
leshields
Relations between the real projective space and canonical covering projections
Let $\displaystyle p:\mathbb{S}^n \rightarrow \mathbb{R}\mathbb{P}^n (n\geq 1)$ be the canonical covering projection. For any map $\displaystyle f: X \rightarrow \mathbb{R}\mathbb{P}^n$ define the space

$\displaystyle f*X = \{(x,y) \in X$x$\displaystyle \mathbb{S}^n | f(x) = p(y) \in \mathbb{R}\mathbb{P}^n\}$.

and the covering projection

$\displaystyle q: f*X \rightarrow X ; (x,y) \mapsto x$

i) What is the fibre of $\displaystyle q$?

ii) For any space $\displaystyle W$ establish a one-one correspondence between maps $\displaystyle g: W \rightarrow f*X$ and pairs of maps $\displaystyle (h:W \rightarrow X, k: W \rightarrow \mathbb{S}^n )$ such that $\displaystyle fh = pk : W \rightarrow \mathbb{R}\mathbb{P}^n$.

iii) Let $\displaystyle \mathbb{Z}_2 = \{1, -1\}$, the space with two points $\displaystyle 1, -1$. For $\displaystyle f = p :X = \mathbb{S}^n \rightarrow \mathbb{R}\mathbb{P}^n$ define a homeomorphism $\displaystyle e : \mathbb{S}^n$x$\displaystyle \mathbb{Z}_2 \rightarrow f*X$ such that

$\displaystyle qe : \mathbb{S}^n$x$\displaystyle \mathbb{Z}_2 \rightarrow \mathbb{S}^n ; (x,y) \mapsto x$ .

I think i) is relatively straightforward and I get for a given $\displaystyle x \in X$ there are two points $\displaystyle a,b \in \mathbb{S}^n$ with $\displaystyle a = -b$ such that $\displaystyle f(x) = p(y) \in \mathbb{R}\mathbb{P}^n$. So $\displaystyle q^{-1}(x) = \{ (x,a), (x,b) \}$.

However I'm struggling with the next two, any help appreciated...