Let M be a smooth manifold. A smooth singular k-simplex of M is a smooth map from the standard simplex in Rk to M. The free abelian group, Sk, generated by singular k-simplices is said to consist of singular k-chains of M. These groups, together with boundary map, ∂, define a chain complex. The corresponding homology (resp. cohomology) is called the smooth singular homology (resp. cohomology) of M.

what a buzzkill, i was hoping it'd be about lollipops, or perhaps snowflakes