The Unapologetic Mathematician

Mathematics for the interested outsider


We can integrate on the standard cube, and on singular cubes in arbitrary manifolds. What if it’s not very easy to describe a region as the image of a singular cube? This is where chains come in.

So a chain is actually pretty simple; it’s just a formal linear combination of singular k-cubes. That is, for each k we build the free abelian group C_k(M) generated by the singular k-cubes in M.

If we have a formal sum c=a_1c_1+\dots+a_lc_l — the c_i are all singular k-cubes and the a_i are all integers — then we define integrals over the chain by linearity:


And that’s all there is to it; just cover the k-dimensional region you’re interested in with singular k-cubes. If there are some overlaps, those areas will get counted twice, so you’ll have to cover them with their own singular k-cubes with negative multipliers to cancel them out. Take all the integrals — by translating each one back to the standard k-cube — and add (or subtract) them up to get the result!

August 5, 2011 Posted by | Differential Topology, Topology | 4 Comments