Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, you need the map to be $C^{\infty}$. This is what bugged me, because using $C^{\infty}$ map from an $n$-simplex, you can "pinch" the boundary at most $n+1$ times. So the image of a 2-simplex can never be, say, a rectangle, or a Koch snowflake. I suppose there should be a better definition of chain for the purpose of integration that does not depend arbitrarily on the shape you started with, but I cannot find it. Can someone explain this to me, or point me to a source?
Thank you.
The best thing to integrate on in a manifold is a submanifold. IIRC the notion of a smooth simplex really only comes up in proving de Rham's theorem: but note you can approximate a rectangle, for instance, as the sum of two smooth 2-simplices, and the Koch snowflake can't possibly be a smooth chain.