Algebraic Topology, Induced Homomorphism

Induced Homomorphism

Let f be a continuous function from the space S into the space T. Assume S and T are path connected. We're going to prove a number of small theorems about homotopies and homotopy classes moving from S to T, culminating in a group homomorphism from π₁(S) into π₁(T). Each of these steps is easy, as long as you remember that the composition of continuous functions is continuous.

If d is a fixed domain, and u(d) and v(d) are homotopic in S, then f(u) and f(v) are homotopic in T. Apply f to the homotopy.
Homotopy classes in S (relative to d) map into homotopy classes in T.
Let d be the unit circle (or a higher dimensional sphere if you like). Loops from d into S, based at a point c, map to loops in T, based at f(c).
Since homotopic loops map to homotopic loops, loop classes in S, based at c, map to loop classes in T, based at f(c). The underlying homotopies, that define the classes, may or may not fix the base point c.
Let d be the unit circle. Let u and v be two loops in S and let w = uv. Apply f to uv and get exactly the same loop as f(u)f(v). Thus f induces a group homomorphism from π₁(S) into π₁(T). The homomorphism need not be onto, even if f is onto, even if f implements a covering space. this is illustrated by wrapping the real line around the circle, a function that carries the trivial group into Z.
The composition of successive continuous functions f and g induces a group homomorphism that is the composition of the homomorphisms induced by f and g.
If f is a homeomorphism, then the induced homomorphism from π₁(S) into π₁(T) is reversible. In other words, π₁(S) = π₁(T). The fundamental group is a topological invariant. If spaces are homeomorphic, they have the same group. This is not surprising, since a homeomorphism merely relabels the points of S, without changing the open sets, so we would not expect loops, or loop classes, or path concatenation (the group operator) to change in any way, and they don't.
If spaces have different fundamental groups they cannot be homeomorphic. The spaces are truly different.
The circle, with fundamental group Z, is different from any convex region of Rⁿ, including Rⁿ itself. In a circle, you can travel around and return to your starting point, with no shortcut, no way to cut through. Rⁿ always has a direct path. That's the intuitive difference; the fundamental group proves our intuition is right.

Category Functor

In the world of category theory, groups and homomorphisms form a concrete category. The same is true of path connected topological spaces and continuous functions. In each case functions carry sets into sets, and obey certain constraints. Homomorphisms respect group operations, while continuous functions pull open sets back to open sets.

The map π₁ is actually a functor between categories. It takes spaces to groups, and the diagram commutes. Apply a continuous function from one space into another, then invoke the functor π₁, and the same could be accomplished by taking π₁ and then applying the corresponding homomorphism.