Wrap the real line around the unit circle via cos(2πx) and sin(2πx). Thus, when x is any integer, we return to our starting point 1,0 on the x axis. Call this function w, since it wraps, or winds, R1 around S1, like a string around a spool. I like to picture the real line above and the circle below, whence w() carries segments upstairs onto arcs downstairs.
Since S1 embeds in the plane, give it the euclidean topology. Sines and cosines are continuous in the plane, hence w is a continuous map from R1 onto S1.
Let p(t) be a closed path from the unit interval I into S1. (I don't mean topologically closed, I mean a closed curve that returns to its starting point.) The path may run around the circle several times, and back up over itself, but since it is a closed path it starts and ends at a point c. In fact, we can sew the two ends of the domain, i.e. the unit interval, together, and our path becomes a continuous image of S1 in S1. conversely, if p carries S1 into S1, we can cut the domain at any point, and p becomes a closed path in S1. Throughout this page we will be talking about closed paths in S1, which lift to paths upstairs in R1, but remember, these are based on the images of S1 in S1. That's what we're interested in after all, the homotopy classes of S1 into S1.
Ok, we have a closed path in S1 that starts and ends at c. Let b be any point in R1 that maps to c. In other words, w(b) = c. The value of b could change by an integer, and only by an integer, if w(b) is to equal c, but for our purposes it doesn't matter what b is, as long as it maps to c.
We are going to lift p(t) to a unique path q(t) in R1, that starts at b, and satisfies w(q(t)) = p(t). This can be done by the unique path lifting theorem, but if you haven't seen that, the idea is straight forward. Take small steps along p(t), and at each step, w is locally invertible. There is no ambiguity; q has to advance or retreat along the real line as p, its shadow downstairs, moves forward and backward around the unit circle. (In this case forward means counterclockwise.)
Since w(q(1)) = c, q(1)-b must be an integer. Let d = q(1)-q(0), which does not depend on our selection of b. Since q is uniquely determined by p, d is a function of p, and is called the degree of p.
Now I call upon another theorem from covering spaces; homotopic functions downstairs have homotopic lifts upstairs. If you don't need to see a rigorous δ ε proof, the reasoning is straightforward. Take small steps in time, as small as you need, and the two paths downstairs, close together in time, are close to each other in space as well. Use a different variable v for the domain of the paths, i.e. p0(v) and p1(v), so that we may use t for time, as the homotopy slides p0 onto p1. Given a sufficiently small interval of time, and a point v in I, pt(v) stays within a small arc of the circle, and w is locally invertible on this arc. The lift, as a function of time, is, and has to be, w inverse applied to pt(v). Do this across the entire homotopy and find a homotopy connecting q0 and q1 upstairs. Also, w applied to the homotopy upstairs gives the homotopy downstairs.
In summary, homotopic paths in S1 produce homotopic paths in R1. Our paths return to start in S1, but this may not be the case in R1. In fact, the difference between the two endpoints is the degree of the path.
Note, the degree measures the number of times p spins around the circle counterclockwise. Moving around clockwise contributes negatively to the degree. This is only an intuitive explanation however. I'm not going to offer a proof here, because p could cross the magical boundary point c an infinite number of times, and that gets a little messy.
Assume p0 and p1 are homotopic closed paths on the circle, or equivalently, they are homotopic images of S1. Let c(t) be pt(0), which traces the motion of the start/end point of our closed path in time. This is a continuous function of t.
For every time t, apply a rotation of the circle that spince c(t) back to c(0), the point we know as c. Verify that this rotation is continuous across time and space, in other words, a homotopy. Since the composition of homotopies is another homotopy, our two paths, with rotations applied, are still homotopic, and as the homotopy carries one path onto the other, the base point c never moves.
Let p0 and p1 have homotopic paths q0 and q1 upstairs. The paths q0 and q1, and every path qt along the way, maps to an integer qt(1)-qt(0), or qt(1)-b, which is the degree of pt. Clearly this map is a continuous function of time, yet the range is a discrete set of points, namely the integers. Therefore the degree of pt remains constant. Two homotoppic functions from S1 into S1 have the same degree.
Conversely, assume p0(I) and p1(I) are closed paths in S1 with the same degree d. Use a homotopy to rotate p1, as we did before, so that p0 and p1 have the same base point c. Now the two paths lift to q0 and q1 in R1, which both start at b and end at b+d. We know that any two paths into real space are homotopic, and the homotopy is linear. Each section of qt expands, or contracts, at a steady rate, until q0 becomes q1. Throughout this process, qt(0) remains constant at b, and qt(1) is always b+d. The ends of the path are nailed down, and do not move with time.
Apply w to this homotopy and find a homotopy that connects p0 and p1 downstairs. Therefore closed paths in S1 are homotopic iff they have the same degree.
Finally, map z onto zd in the complex plane to realize a closed path of degree d for each integer d. This wraps the circle around itself d times, and is valid for any d. The homotopy classes from S1 into S1 correspond 1-1 with the integers.
This would be the case if u maps to any point on the circle other than c. We are simply spinning around the circle too fast to settle down at any one point.
The same reasoning holds for finitely many circles joined at the base point c. Suppose a path loops around one circle 19 times, then around another circle -22 times, then around another circle 12 times, and so on, such that the cumulutive sum of the absolute value of these degrees is unbounded. Find u as before, the point where the path must race around the circles at an infinite speed. The smallest circle has radius r, and points arbitrarily close to u have images that are at least r units away from f(u), whence the path is not continuous at u. we'll talk more about the homotopy classes of the generalized figure 8 in subsequent sections. For now, the degree is always finite, and we have characterized the homotopy classes of the circle.