## 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 -cubes. That is, for each we build the free abelian group generated by the singular -cubes in .

If we have a formal sum — the are all singular -cubes and the are all integers — then we define integrals over the chain by linearity:

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

[…] 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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[…] 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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[…] 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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[…] 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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