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

Stub grade: B

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Better title needed!

Statement

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

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

then:

• $(\varphi\circ f_1)\simeq(\varphi\circ f_2)\ (\text{rel }\{0,1\})$

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

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

Define:

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

Grade: C

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

It remains to be shown that $H'$ is a homotopy between $(\varphi\circ f_1)$ and $(\varphi\circ f_2)$ 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