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?


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).

  • $\begingroup$ Thanks. And also, one can speak about based homotopy only when having based maps? $\endgroup$ – edward_scissorhands Mar 6 '17 at 17:39
  • 1
    $\begingroup$ That's correct. $\endgroup$ – Michael Albanese Mar 6 '17 at 17:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.