# The Unapologetic Mathematician

## Chains

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:

$\displaystyle\int\limits_c\omega=a_1\int\limits_{c_1}\omega+\dots+a_l\int\limits_{c_l}\omega$

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