# Null homologous loop and orientable surface

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.

• they must also mean compact surface – Tim kinsella Oct 7 '18 at 5:07
• 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. – Nick L Oct 7 '18 at 9:39