# 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!

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Posted by John Armstrong | Differential Topology, Topology

## 4 Comments	»

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

Pingback by | August 9, 2011 | Reply

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 [...]

Pingback by | August 17, 2011 | Reply

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

Pingback by | September 5, 2011 | Reply

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 [...]

Pingback by | September 8, 2011 | Reply

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## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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