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!

August 5, 2011 Posted by | Differential Topology, Topology | 4 Comments

   

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