Let $X$ be a topological space and $A$ a retract of $X$. Is the fundamental group of $A$ a subgroup of the fundamental group of $X$?
2026-04-03 08:14:20.1775204060
Is the fundamental group of a retract a subgroup of the original space?
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Let $i$ be the inclusion and $r$ the retraction, so that $r\circ i$ is the identity. Then $(r\circ i)_*=r_*\circ i_*$ is the identity. Thus $r_*$ is surjective and $i_*$ is injective. From the latter fact we deduce that the fundamental group of the retract is isomorphic to a subgroup of the fundamental group of the larger space.