Algebraic Topology, The Fundamental Group

The Fundamental Group

In the last section we associated a set, the set of integers, with the topological space S¹, i.e. the unit circle. Now the set of integers is not, by itself, an algebraic structure. Algebra usually entails groups, or rings, or fields, or something along those lines. So we haven't quite reached the land of "algebraic topology" yet, but we're about to.

In this section we will introduce a group operator, turning the set of integers into the group Z. In fact, Z is the fundamental group of S¹, an idea first developed by Poincare. (biography)

Let S be a path connected space and let c be a base point in S. Let a loop be the continuous image of a fixed circle (the domain), into S (the range), such that the base point of the circle maps to the base point c in S. Two loops are homotopic if there is a homotopy between them that fixes c. In other words, c does not move as one loop slides onto the other. The set of homotopy classes of loops will become the fundamental group for the space S. (We'll prove this does not depend on c later.)

Two loops are combined by concatenation. Map the first half of the circle onto the first loop, and the second half of the circle onto the second loop. Since the image is continuous, and starts and ends at c, the result is another loop.

We need to show this is a well defined operator on homotopy classes. Assume f₀ and f₁ are homotopic (the first loops), and g₀ and g₁ are homotopic ( the second loops). Run the first homotopy twice as fast, carrying f₀ onto f₁ and leaving g₀ alone. Then run the second homotopy twice as fast, carrying g₀ onto g₁ and leaving f₁ alone. this builds a homotopy from f₀*g₀ to f₁*g₁, and concatenation, or *, is a valid operator on homotopy classes of loops.

To show asssociativity, note that (fg)h is a reparameterization of f(gh). In the first expression, f and g are images of the first and second quarter of the circle, and h is the image of the second half of the circle. In the second expression, g and h are images of the third and fourth quarter of the circle, and f is the image of the first half of the circle. Apply a linear homotopy to the domain, that pulls the circle around, so that f g and h each consume a third of the circle. Thus the loops are homotopic, and concatenation is associative.

The identity class is represented by the constant function c. Concatenate this with some other path f, and f is the image of the first half of the circle, while the second half maps to c. Expand the first half of the circle linearly while the second half shrinks to a point, and find a homotopy that takes you back to f. Concatenation with the constant function c does not change the homotopy class; this must be the identity element.

The inverse of f(x) is f(1-x), running the loop in the opposite direction. Concatenate f with its inverse, and the composite function g now runs around the loop, stops at c, and retraces the loop. We're going to pull g back, so that it runs partway around the loop, sits there for a while, then retraces its path back to c. Pull the stopping point back linearly with time, until, at the end of the homotopy, g never leaves c at all. Hence f concatenated with its inverse produces the identity element of the group.

We now have a group of homotopy classes, which is the fundamental group of S. This is written π₁(S). Higher order functors exist, written π_n(S). These are based on the homotopy classes of Sⁿ (the n sphere) into S. These are not dealt with here.

Base Point Invariance

Let d be another point in S, defining another fundamental group H. Map G into H as follows. Given a loop at c, draw a path from d to c, run the loop, and retrace the path from c back to d. This is a loop at d, and a homotopy between loops at c can be used to build a homotopy between the corresponding loops at d. (Just hold the paths between c and d constant.) Thus the map from G into H is well defined.

Let e and f be two loops based at c. Of course ef is another loop at c, and another element of the group G. Take the corresponding loops based at d and concatenate them. The path starts at d, runs to c, traces the loop e, goes back to d, then back to c, then around the loop f, then back to d. The middle step, from c to d to c, can be retracted by a homotopy, until we just sit at c. Then the time spent sitting on c can be compressed to nothing, leaving the loop ef, relative to the base point d. In other words, our map respects concatenation, and is a group homomorphism from G into H.

By symmetry, another homomorphism maps H into G. Let's compose them. The loop e is based at c; change the base to d, and then back to c. The new loop runs from c to d to c, around e, then from c to d to c. The excursions to d can be retracted by a homotopy, leaving only the loop around e. The homomorphisms are inverses of each other, hence G and H are isomorphic.

In summary, the fundamental group does not depend on c.

The Fundamental Group of the Circle

In the previous section we showed the homotopy classes of S¹ into S¹ correspond to the integers, as determined by the degree of the path. We need to show that loop concatenation corresponds to addition in the integers. When two paths are concatenated, their lifts in R¹ are concatenated as well. The difference between end points, which is the degree after all, is the difference in end points of the first lift plus the difference in end points of the second. In other words, the degree of the sum equals the sum of the degrees, and the fundamental group of S¹, written π₁(S¹), = Z.