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.

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):

${\displaystyle {\begin{aligned}h^{*}([f]):=[h\circ f]\end{aligned}}}$

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

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