Fundamental group homomorphism induced by a continuous map

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Important page, although the proof is done

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]p\in X[/ilmath] be some fixed point (to act as the base point for the fundamental group, [ilmath]\pi_1(X,p)[/ilmath]) and let [ilmath]\varphi:X\rightarrow Y[/ilmath] be a continuous map. Then:

• [ilmath]\varphi[/ilmath] induces a group homomorphism on the fundamental groups, [ilmath]\pi_1(X,p)[/ilmath] to [ilmath]\pi_1(Y,\varphi(p))[/ilmath], which we denote:
  • [ilmath]\varphi_*:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))[/ilmath] defined as:
  • [ilmath]\varphi_*:[f]\mapsto [\varphi\circ f][/ilmath]

For more details on the formalities of the definition see the proof: the proof, which names everything involved during the statement.

Immediate results

The induced homomorphism of a composition is the same as the composition of induced homomorphisms

Let [ilmath](X,\mathcal{ J })[/ilmath], [ilmath](Y,\mathcal{ K })[/ilmath] and [ilmath](Z,\mathcal{ H })[/ilmath] be topological spaces, let [ilmath]p\in X[/ilmath] be any fixed point (to act as a base point for the fundamental group [ilmath]\pi_1(X,p)[/ilmath]) and let [ilmath]\varphi:X\rightarrow Y[/ilmath] and [ilmath]\psi:Y\rightarrow Z[/ilmath] be continuous maps. Then:

• [ilmath](\psi\circ\varphi)_*\eq(\psi_*\circ\varphi_*)[/ilmath]
  • where [ilmath]\varphi_*[/ilmath] denotes the fundamental group homomorphism, [ilmath]\varphi_*:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))[/ilmath], induced by [ilmath]\varphi[/ilmath] - and "" for the others

Note that both of these maps have the form [ilmath]\big(:\pi_1(X,p)\rightarrow\pi_1(Z,\psi(\varphi(p))\big)[/ilmath]

The induced homomorphism of the identity map is the identity map of the fundamental group

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, let [ilmath]\text{Id}_X:X\rightarrow X[/ilmath] be the identity map, given by [ilmath]\text{Id}_X:x\mapsto x[/ilmath] and let [ilmath]p\in X[/ilmath] be given (this will be the basepoint of [ilmath]\pi_1(X,p)[/ilmath]) then:

• the induced map on the fundamental group [ilmath]\pi_1(X,p)[/ilmath] is equal to the identity map on [ilmath]\pi_1(X,p)[/ilmath]
  • That is to say [ilmath](\text{Id}_X)_*\eq\text{Id}_{\pi_1(X,p)}:\pi_1(X,p)\rightarrow\pi_1(X,p)[/ilmath] where [ilmath]\text{Id}_{\pi_1(X,p)} [/ilmath] is given by [ilmath]\text{Id}_{\pi_1(X,p)}:[f]\mapsto [f][/ilmath]

Homeomorphic topological spaces have isomorphic fundamental groups

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be homeomorphic topological spaces, let [ilmath]p\in X[/ilmath] be given (this will be the base point of the fundamental group [ilmath]\pi_1(X,p)[/ilmath]) and let [ilmath]\varphi:X\rightarrow Y[/ilmath] be that homeomorphism. Then: :

• [ilmath]\pi_1(X,p)\cong\pi_1(Y,\varphi(p))[/ilmath] - where [ilmath]\cong[/ilmath] denotes group isomorphism here, but can also be used to denote topological isomorphism (AKA: homeomorphism)

That is to say:

• [ilmath]\big(X\cong_\varphi Y)\implies(\pi_1(X,p)\cong_{\varphi_*}\pi_1(Y,\varphi(p))\big)[/ilmath]

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
  • Retract - these have very useful properties
  • Deformation retract
    • Strong deformation retract