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A surface $S$ is orientable iff its first homology group $H_1(S)$ is a free abelian group $F$, or it is non-orientable iff $H_1(S)$ has the form $F + \mathbb{Z}_2$ (so says wiki, "orientability and homology" header).

My question: are there similarly nice geometric interpretations for the structure of higher-order homology groups? For example, if $H_2(S)$ is free abelian, is there some intuitive sense in which we can say that $S$ is "second-order orientable" (whatever that means)?

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    That characterization of orientability in terms of H1 is a red herring, really. Orientability is related to the top (co)homology of a manifold. – 2017-02-06

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