Is the winding number of a map $\Omega: \; S^n \mapsto S^n$ dependent on the radius of both spheres? I have 2 questions regarding homotopy groups. 
My first questions comes from the fact that I've read different definitions of homotopy groups. The book by Manton and Sutcliffe defines them as: 
''The set of homotopy classes of based maps $\Psi: S^n \mapsto Y$'', where $S^n$ is a n-dimensional unit sphere. But most other sources that I've read about about this topic do not define $S^n$ necessary as a unit sphere, they don't define the radius of the sphere at all, therefore I'm not sure what is right or wrong.
My second question is for the situation where we consider a mapping from $\Psi: S^n \mapsto S^n$, i.e. $\pi_n(S^n)=\mathbb{Z} \; \forall \; n$. Now I've read in a lot of sources (for instance in Magnetic Monopoles by John Preskill) that say the winding number is a conserved quantity. I understand that the winding number is "preserved by continuous deformations''. 
However, I wonder if the winding number is also conserved if the radius of either n-spheres is changing with time. For instance, there exist a situation where $\pi_1(S^1)=2$, but there is surely also a situation where $\pi_1(S^1)=3$. I.e. they have different winding numbers, in my mind (I'm most likely wrong) this is because the radius of either spheres have changed. I don't, intuitively, understand how else this is possible.
Just in case my background is important: I'm physics student, so have no mathematical skills regarding topology. I'm not very good in mathematical proofs, but I would really like to intuitively understand this topic.
Thanks in advance!
 A: The radius of the sphere doesn't matter. Moreover, if you take a map from a particular $n$-sphere to $\mathbb R^{n+1}-\{0\}$, then you really see it as a winding number around the origin. As long as you never hit the origin by mistake as $t$ varies, this number will stay constant.
A: $\pi_1(S^1) \simeq \mathbb{Z}$ means that the fundamental group of the circle is isomorphic to the entire group of the integers, so it does not make sense to write $\pi_1(S^1) = 2$ or $\pi_1(S^1) = 3$.
Perhaps you mean that there could be two maps $f$ and $g: S^1 \to S^1$ such that the winding number of $f$ is $2$ and the winding number of $g$ is $3$.  This can definitely happen.  Think of the unit circle as being parametrized by radians from $0$ to $2\pi$ with $0 = 2\pi$.  (In topology, we'd say that the circle is homeomorphic to the interval $[0,2\pi]$ with its ends identified.)  The map $\theta \mapsto 2\theta$ has winding number $2$, while the map $\theta \mapsto 3\theta$ has winding number $3$.  If you prefer to think of the unit circle as sitting in $\mathbb{R}^2$, think about the maps $\theta \mapsto e^{2i\theta}$ and $\theta \mapsto e^{3i\theta}$.
So radius doesn't play a role here.
One more thing: algebraic topologists generally don't think of $S^n$ as sitting inside $\mathbb{R}^{n+1}$ -- they prefer a perspective like "$[0,2\pi]$ with the ends identified."  Because of that, they call tend to call winding number "degree."  I tell you this only because it might be helpful in finding out more from mathematicians -- I defer to your judgment about the correct term to use in physics!
