Induced homomorphism (fundamental group)

In mathematics, especially in the area of topology known as algebraic topology, the induced homomorphism is a group homomorphism related to the study of the fundamental group.

Definition

Let X and Y be topological spaces, x₀ ∈ X, y₀ ∈ Y, and let h:X→Y be a continuous map such that h(x₀) = y₀. Define a map h^∗ from π₁(X, x₀) to π₁(Y, y₀) by composing a loop in π₁(X, x₀) with h to get a loop in π₁(Y, y₀):

[math]\displaystyle{ \begin{align} h^*([f]):= [h\circ f] \end{align} }[/math]

(where [math]\displaystyle{ [f] }[/math] denotes the equivalence class of the loop [math]\displaystyle{ f }[/math] under the homotopy relation). Then h^∗ is a homomorphism between fundamental groups known as the homomorphism induced by h.

• If f is a loop in π₁(X, x₀), then h^∗(f) is a loop in π₁(Y, y₀), namely h^∗(f) is a continuous map from [0, 1] to Y, and h^∗(f(0)) = h^∗(x₀) = y₀ and h^∗(f(1)) = h^∗(x₀) = y₀.
• h* is a homomorphism. To avoid repetition, whenever we call f and g loops, they will be known as loops based at x₀. Let f and g be two loops, • be the group operation on π₁(X, x₀) and + be the group operation on π₁(Y, y₀),

h^∗(f • g) = h^∗(f(2t)) for t in [0, 1/2] = (h^∗(f)) + (h^∗(g))

h^∗(f • g) = h^∗(g(2t − 1)) for t in [1/2, 1] = (h^∗(f)) + (h^∗(g))

so that h^∗ is indeed a homomorphism.

• Checking h^∗ is a function (i.e. every loop in π₁(X, x₀) gets mapped onto a unique loop in π₁(Y, y₀)) follows from the fact that if f and g are loops in π₁(X, x₀) that are homotopic via the homotopy H, then h^∗(f) and h^∗(g) are homotopic via the homotopy h^∗H.

Theorem

Suppose X and Y are two homeomorphic topological spaces. If h is a homeomorphism from X to Y, then the induced homomorphism, h^∗ is an isomorphism between fundamental groups [where the fundamental groups are π₁(X, x₀) and π₁(Y, y₀) with h(x₀) = y₀].

Proof

It has already been checked in note 2 that h^∗ is a homomorphism. It remains to check that h^∗ is bijective. If p is the inverse of h, then p^∗ is the inverse of h^∗. This follows from the fact that (p(h))^∗(f) = p^∗(h^∗(f)) = f = (h(p))^∗(f) = h^∗(p^∗(f)). If f and g are two loops in X where f is not homotopic to g, then h^∗(f) is not homotopic to h^∗(g); if F is a homotopy between them, p^∗(F) would be a homotopy between f and g. If k is any loop in π₁(Y, y₀), then h^∗(p^∗(k)) = k, where p^∗(k) is a loop in X. This shows that h^∗ is bijective.

Applications of the theorem

1. The torus is not homeomorphic to R² for their fundamental groups are not isomorphic (their fundamental groups don’t have the same cardinality). A simply connected space cannot be homeomorphic to a non-simply connected space; one has a trivial fundamental group and the other does not.

2. Any two topological spaces have homomorphic fundamental groups (at a particular base point). See note 2 where h^∗ is the homomorphism induced by the constant map. However, they need not have isomorphic fundamental groups (at a particular base point). This shows that the fundamental groups of any two topological spaces always have the same 'group structure'.

3. The fundamental group of the unit circle is isomorphic to the group of integers. Therefore, the one-point compactification of R has a fundamental group isomorphic to the group of integers (since the one-point compactification of R is homeomorphic to the unit circle). This also shows that the one-point compactification of a simply connected space need not be simply connected.

4. The converse of the theorem need not hold. For example, R² and R³ have isomorphic fundamental groups but are still not homeomorphic. Their fundamental groups are isomorphic because each space is simply connected. However, the two spaces cannot be homeomorphic because deleting a point from R² leaves a non-simply connected space but deleting a point from R³ leaves a simply connected space (If we delete a line lying in R³, the space wouldn’t be simply connected any more. In fact this generalizes to Rⁿ whereby deleting a (n − 2)-dimensional parallelepiped from Rⁿ leaves a non-simply connected space).

5. If A is a strong deformation retract of a topological space X, then the inclusion map from A to X yields an isomorphism between fundamental groups.

See also

• Induced homomorphism (Algebraic topology)
• Induced homomorphism (Category theory)

References

• James Munkres (1999). Topology, 2nd edition, Prentice Hall. ISBN:0-13-181629-2.