# Question on concept of homology in calibrated geometry

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?

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.
• Sorry, I think I should have been more specific. To simplify, let us assume that $M$ and $M'$ have no boundary. The proof I know assumes that there is a smooth submanifold $N$ such that $\partial(N)=M-M'$. Then we can use Stokes theorem: $$\int_M \phi-\int_{M'} \phi=\int_{\partial N} \phi=0.$$ I don't know how to infer this without assuming such an N. If $M$ and $M'$ are homologous in singular homology, there is a singular simplex (or a sum of them) whose boundary is $M-M'$. But this needn't be a smooth submanifold. This is the point I don't understand. Oct 23, 2019 at 8:37