Notes:Homotopy terminology

Homotopy

Homotopy is a continuous map, $F:X\times I\rightarrow Y$ where $I$ denotes the unit interval, $[0,1]\subseteq\mathbb{R}$.
Here $X$ and $Y$ are topological spaces

Homotopic maps

If $f,g:X\rightarrow Y$ are continuous maps, we say that "$f$ is homotopic to $g$" if:

• There is a homotopy, $F:X\times I\rightarrow Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ ($\forall x\in X$)

Homotopic relative to $A$

If $f,g:X\rightarrow Y$ are continuous maps and $A\in\mathcal{P}(X)$, we say that "$f$ is homotopic to $g$ relative to $A$" if:

• There is a homotopy, $F:X\times I\rightarrow Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ AND
• $F(a,t)=f(a)=g(a)$ for all $t\in I$ and $\forall a\in A$

Homotopic paths

This is a special case, here we are dealing with $A:=\{0,1\}$ and $F:I\times I\rightarrow X$, the maps we are building a homotopy between are of the form:

• $\alpha:I\rightarrow X$

And we say:

• $\alpha_1$ is homotopic to $\alpha_2$ (rel $\{0,1\}$) if there is a homotopy between them.