Let $M$ and $N$ be connected (compact if you like) manifolds and let $x \in M$ and $y \in N$ be specified base points. Suppose that $f,g : M \to N$ with $f(x) = g(x) = y$ and suppose that $f$ and $g$ are homotopic. Does it follow that $f$ and $g$ are homotopic relative to $\{x\}$?
Based homotopy equivalent vs homotopy equivalent for manifolds?
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1 Answers
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No. For $M = S^1$, $N = $ some surface, this is the same as asking if homotopy of loops is the same as free homotopy of loops in a surface. They are not. In the picture below, the red and green curves are freely homotopic (i.e., if you let the blue point move on one buyt not the other) but not homotopic rel the blue point. (Note that part of the green curve is dotted, indicating it is running around behind the back of the two-holed torus; sorry the dots don't show up well).
