Consider $\displaystyle S^m$ embedded in $\displaystyle S^n (m < n) $ as the subspace $\displaystyle \{(x_1,x_2,...,x_{m+1},0,....,0) | {\sum x_i^2 = 1}\}$. Show $\displaystyle S^n$ \ $\displaystyle S^m$ is homotopy equivalent to $\displaystyle S^{n-m-1}$.

If we consider m=1, n=2, then according to the theorem we're trying to prove, our quotient space should be homotopy equivalent to a point. However from our definition, $\displaystyle S^1$ embedded in $\displaystyle S^2$ is $\displaystyle \{(x_1,x_2,0) | {\sum x_i^2 = 1}\}$, i.e. the circle on the x-y plane. I thought the quotient space meant "shrinking all the points in the same equivalence class into a point", so our quotient space ends up more like $\displaystyle S^2 \wedge S^2$? So my first question is, is the question giving the right definition of embedding for us to prove the proposition?

In the case of m=1, n=2, if we use an embedding definition based on rotation about the z-axis (e.g. using spherical polars, points are in the same equivalence class if they have the same radius and elevation angle from x-y plane), we can see very clearly that $\displaystyle S^2$ \ $\displaystyle S^1$ is [-1,1]. Even if the embedding definition is correct in question, maybe there is an easier one to work with?

I probably have completely misunderstood the question, would love to know my mistake. Any comments are welcome.