The Unapologetic Mathematician

Mathematics for the interested outsider


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:


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. […]

    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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