Does there exists a space X, along with two loops $f,g$ based at $x_0 \in X$ such that $f$ and $g$ are freely homotopic and not "based point" homotopic?

Two loops are freely homotopic if there exists a homotopy between.

A base point preserving homotopy is one where the homotopy between $f$ and $g$ also satisfies $F(x_0,t) = x_0$


Yes there is. In general, the set of homotopy classes of maps $S^1 \to X$ corresponds to the conjugacy classes of $\pi_1(X, x_0)$. So any two elements of $\pi_1(X, x_0)$ which are conjugate but not equal will provide such an example; note, for such elements to exist, $\pi_1(X, x_0)$ must be non-abelian.

It's not a great picture, but the blue and red loops below are freely homotopic, but not homotopic relative to the basepoint $x_0$. In $\pi_1(X, x_0)$, they are conjugate via the green loop.

enter image description here

  • 1
    $\begingroup$ You are beautiful. $\endgroup$ – Dionel Jaime Oct 5 '17 at 3:24

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.