The Unapologetic Mathematician

Mathematics for the interested outsider

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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August 5, 2011 - Posted by | 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 Cubic Singular Homology « The Unapologetic Mathematician | 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 Stokes’ Theorem (statement) « The Unapologetic Mathematician | 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 Integrals over Manifolds (part 1) « The Unapologetic Mathematician | 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 Switching Orientations « The Unapologetic Mathematician | September 8, 2011 | Reply


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