I am reading Algebraic Topology: A First Course written by Greenberg and Harper. On page 67 of this book it is stated that
Let $\gamma$ be a loop in $X$ regarded as a map $f:S^1\to X$. For $\chi[\gamma] =0$ it is necessary and sufficient that $f$ extend to $\tilde f:W\to X$ where $W$ is an orientable surface with boundary $S^1$.
After that the authors sketch the proof of this theorem. I have not understood what do they mean by orientability, is it same as what we do in differential topology- orientability of a manifold.
Any reference or book or article containing this type theorem and proof will be helpful to me.
I also need meterial which contains solid presentation on orientable surface in algebraic topology.