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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.

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  • $\begingroup$ they must also mean compact surface $\endgroup$ – Tim kinsella Oct 7 '18 at 5:07
  • $\begingroup$ It is the notion of orientability of manifold that we are familiar with. A manifold with boundary is called orientable if the complement of the boundary has an orientation. $\endgroup$ – Nick L Oct 7 '18 at 9:39

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