Induced Homomorphisms

Algebraic Topology · Axiom Academy

How continuous maps between spaces induce group homomorphisms between fundamental groups Let f: (X, x₀) → (Y, y₀) be a continuous map between pointed spaces. Given a loop γ in X based at x₀, we can compose it with f to get a loop in Y: 2. Well-Defined on Homotopy Classes We need to verify that f₊ is well-defined, meaning it doesn't depend on which representative of the homotopy class we choose. f is continuous, so composing with f preserves homotopies If H: [0,1] × [0,1] → X is a homotopy, then f ∘ H: [0,1] × [0,1] → Y is also continuous The basepoint is preserved: f(x₀) = y₀ throughout The map f₊ respects the group operation (concatenation of loops). To see this, let [γ₁] and [γ₂] be elements of π₁(X, x₀): The key insight: composing f with a concatenated path gives the concatenation of the composed paths! The induced homomorphism construction has powerful applications: Constant map: If f maps all of X to a single point y₀, then f₊ is the trivial homomorphism (everything maps to the identity) Inclusion: If A ⊆ X and i: A → X is inclusion, then i₊: π₁(A) → π₁(X) tells us which loops in A are also contractible in X Homeomorphism: If f is a homeomorphism, then f₊ is an isomorphism (we'll explore this more when studying functoriality)

This is the written version of the interactive lesson above. See the full Algebraic Topology course.