# Topology question

• Nov 23rd 2008, 10:41 AM
TheEmptySet
Topology question
I am looking for a surjective map

$\displaystyle g:\{S^{n-1}\times [-1,1] \} \to S^{n}$

Where $\displaystyle S^n=\{ x \in \mathbb{R}^{n+1}:||x||=1 \}$

I have figured out the case for n=1(The easy one)

It is basically converting to sphereical coordinates.

I can't find a closed form for it(I'm not even sure if one exists) for general n.

Just for context I'm trying to show that the suspension of $\displaystyle S^{n-1}$ is homeomorphic to $\displaystyle S^n$

So I need the function g above to be a quotient map.

Thanks for any input
• Nov 23rd 2008, 11:38 AM
TheEmptySet
let $\displaystyle \vec x \in S^{n-1}$ and $\displaystyle t\in [-1,1]$

What I have been thinking is this (I know it is not a function)

$\displaystyle g(\vec x, t)=(h(\vec x),1-t^2)$

Where $\displaystyle h( \vec x,t) =\{x_1^2+x_2^2+...x_{n-1}^2=1-t^2 \}$

The problem with this is it returns a set and is not well defined. I guess what I'm not sure about is it that if I require each $\displaystyle x_i$ from the input to have the same sign in the output would this fix the defect?

My intution says yes, but then again I have been working on this for a while and maybe I just want it to be right?

Thanks again

TES
• Nov 23rd 2008, 12:43 PM
Opalg
$\displaystyle g(\vec x, t)=((1-t^2)^{1/2}\vec x,t)$
• Nov 23rd 2008, 02:57 PM
TheEmptySet
Quote:

Originally Posted by Opalg
$\displaystyle g(\vec x, t)=((1-t^2)^{1/2}\vec x,t)$

Thank you.