# 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 - Posted by | Differential Topology, Topology

1. [...] that we’re armed with chains — formal sums — of singular cubes we can use them to come up with a homology theory. [...]

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2. [...] anyway, on to the theorem! We know how to integrate a differential -form over a -chain . We also have a differential operator on differential forms [...]

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3. [...] defined how to integrate forms over chains made up of singular cubes, but we still haven’t really defined integration on manifolds. [...]

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4. [...] sensible to identify an orientation-preserving singular cube with its image. When we write out a chain, a positive multiplier has the sense of counting a point in the domain more than once, while a [...]

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