Wow, it took me so long to find out what is wrong with this - I thought I was taking a quotient space, but I actually need to remove the points of the embedding, and then can explicitly construct a homotopy. Well that was dumb of me....
Consider embedded in as the subspace . Show \ is homotopy equivalent to .
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, embedded in is , 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 ? 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 \ 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.
Wow, it took me so long to find out what is wrong with this - I thought I was taking a quotient space, but I actually need to remove the points of the embedding, and then can explicitly construct a homotopy. Well that was dumb of me....