Question on concept of homology in calibrated geometry

Question

The fundamental lemma of calibrated geometry states that calibrated submanifolds are absolutely volume minimising in their homology class. In the proof, homology equivalent is used synonymously with cobordant. With respect to which homology theory are they minimising? Are they volume minimising with respect to singular homology?

Answer 1

I don't know what proof you're referring to that uses cobordant. The proof I know uses singular homology (say with piecewise-smooth chains) to deduce that we have $\int_M \phi = \int_{M'}\phi$ when $\partial M=\partial M'$, $M$ is homologous to $M'$, and $\phi$ is closed.