The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths

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Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]\varphi:X\rightarrow Y[/ilmath] be a continuous map and let [ilmath]f_1,f_2:I\rightarrow X[/ilmath] be paths in [ilmath]X[/ilmath] such that:

• [ilmath]f_1\simeq f_2\ (\text{rel }\{0,1\})[/ilmath] (that is to say [ilmath]f_1[/ilmath] and [ilmath]f_2[/ilmath] are end-point-preseriving homotopic)

then:

• [ilmath](\varphi\circ f_1)\simeq(\varphi\circ f_2)\ (\text{rel }\{0,1\})[/ilmath]

That is to say:

• The relation of paths being end-point-preseriving-homotopic is preserved under composition with continuous maps.

Purpose

This is a precursor theorem to:

• A continuous map induces a homomorphism on fundamental groups

Proof

We need to show that [ilmath]H':\ (\varphi\circ f_1)\simeq(\varphi\circ f_2)\ (\text{rel }\{0,1\})[/ilmath], we will do this by constructing a homotopy, [ilmath]H':I\times I\rightarrow Y[/ilmath], between them. We know that:

• [ilmath]H:f_1\simeq f_2\ (\text{rel }\{0,1\})[/ilmath], where [ilmath]H:I\times I\rightarrow X[/ilmath] is the homotopy between the paths [ilmath]f_1[/ilmath] and [ilmath]f_2[/ilmath].

Define:

• [ilmath]H':I\times I\rightarrow Y[/ilmath] by [ilmath]H':(u,t)\mapsto \varphi(H(u,t))[/ilmath], or more explicitly: [ilmath]H':\eq\varphi\circ H[/ilmath]
  • This is continuous as the composition of continuous maps is continuous

Grade: C

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It remains to be shown that [ilmath]H'[/ilmath] is a homotopy between [ilmath](\varphi\circ f_1)[/ilmath] and [ilmath](\varphi\circ f_2)[/ilmath] we must show that the initial and final stage of the homotopy is equal to them, this is really not difficult!

See also

• A continuous map induces a homomorphism on fundamental groups