Let $f, g$ be two paths in a space such that their concatenation $f * g$ is nullhomotopic. Prove that $f$ is homotopic to $g$ rel $\{0,1\}$.
Homotopy of two paths whose composition is nullhomotopic
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algebraic-topology
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0You can't prove it, because it's not true. – 2010-12-14
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1To elabore on Ryan's comment: Your statement becomes ture when you have to replace $g$ with its reversed path. – 2010-12-14