I have a question concerning the right understanding of the following two definitions: free and based homotopy.
If I am right, free homotopy between two continuous maps $f, g : X \to Y$ is a continuous map $F: X \times I \to Y$ such that
\begin{align*}
F(x,0) &= f(x),\ \ \forall x\in X\\
F(x,1) &= g(x),\ \ \forall x \in X
\end{align*}
(with no extra condition that these two maps have to be based maps?)
The other definition (based homotopy) would be the following:
For two based maps $f,g: (X, x_0) \to (Y, y_0)$ one says that they are based homotopic if there exists a continuous map $G : X \times I \to Y$ such that
\begin{align*}
G(x,0) &= f(x),\ \ \forall x\in X\\
G(x,1) &= g(x),\ \ \forall x \in X\\
G(x_0,t) &= y_0,\ \ \forall t \in I
\end{align*}
Does the second definition imply that we can speak only about based homotopy when having based maps?
Best Answer
This is correct.
You can talk about free homotopies between any two maps $f, g : X \to Y$. If $f$ and $g$ are based maps, you can still talk about free homotopies between them, you just ignore the fact that they are based maps (in particular, the basepoint plays no role).
You can only talk about based homotopies between based maps (otherwise the third condition makes no sense).