---- > [!theorem] Theorem. ([[deformation retraction preserves the fundamental group]]) > Let $X$ be a [[topological space]]. Let $A$ be a [[deformation retract]] of $X$; let $x_{0} \in A$. Then the [[inclusion map]] $\iota: (A,x_{0}) \to (X,x_{0})$ > [[homomorphism of fundamental groups induced by a continuous map|induces]] an [[group isomorphism|isomorphism]] of [[fundamental group|fundamental groups]]. > [!proof]- Proof. ([[deformation retraction preserves the fundamental group]]) > This is an immediate generalization of [[punctured Euclidean space and the unit sphere have isomorphic fundamental groups|the proof here]]. In that case, we explicitly constructed $r$ as the normalization function $v \mapsto \frac{v}{\|v\|}$. Here, we are given that $r$ exists with $\iota \circ r$ homotopic to $\id$ as part of the [[deformation retract]] definition. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag