Fundamental group homomorphism induced by a continuous map

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Important page, although the proof is done

Definition

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

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

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

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

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

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

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

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

Homeomorphic topological spaces have isomorphic fundamental groups

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

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

That is to say:

• $\big(X\cong_\varphi Y)\implies(\pi_1(X,p)\cong_{\varphi_*}\pi_1(Y,\varphi(p))\big)$

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