View source for Fundamental group homomorphism induced by a continuous map

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==Definition==
Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]<ref group="Note">Of course (and as usual) there is no reason why these cannot be the same spaces</ref>, let {{M|p\in X}} be some fixed point (to act as the base point for [[the fundamental group]], {{M|\pi_1(X,p)}}) and let {{M|\varphi:X\rightarrow Y}} be a [[continuous map]]. Then{{rITTMJML}}:
* {{M|\varphi}} induces a [[group homomorphism]] on the fundamental groups, {{M|\pi_1(X,p)}} to {{M|\pi_1(Y,\varphi(p))}}, which we denote:
** {{M|\varphi_*:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))}} defined as:
** {{M|\varphi_*:[f]\mapsto [\varphi\circ f]}}<ref group="Note">{{Caveat|Remember that {{M|[f]}} means {{M|f}} represents the [[equivalence class]] - this definition must be "[[well-defined]]" for whichever {{M|f}} we use.}} As such we must check that for {{M|\alpha\in\pi_1(X,p)}} that for any {{M|f}} and {{M|g}} such that {{M|f,g\in\alpha}} that {{M|[\varphi\circ f]\eq[\varphi\circ g]}}</ref>

For more details on the formalities of the definition see the proof: [[A continuous map induces a homomorphism on fundamental groups#Statement|the proof]], which names everything involved during the statement. 
==Immediate results==
===[[The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms|The induced homomorphism of a composition is the same as the composition of induced homomorphisms]]===
{{:The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms/Statement}}
===[[The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group|The induced homomorphism of the identity map is the identity map of the fundamental group]]===
{{:The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group/Statement}}
===[[Homeomorphic topological spaces have isomorphic fundamental groups]]===
{{:Homeomorphic topological spaces have isomorphic fundamental groups/Statement}}
==Proof==
* Refactored into own page, see: [[A continuous map induces a homomorphism on fundamental groups]]
==See also==
* [[The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms]]
* [[The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group]]
** [[Homeomorphic topological spaces have isomorphic fundamental groups]] (a corollary to the previous claim)
* [[Types of topological retractions]]
** {{link|Retract|topology}} - these have very useful properties
** [[Deformation retract]]
*** [[Strong deformation retract]]

==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology|Algebraic Topology|Homotopy Theory}}