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Suppose c is a k+1 chain in U(open set in space R^n), then boundary of c (a k chain) can be expressed as a linear combination of k-cubes, using boundary operator: $$∂c=∑_ia_ic_i$$, where a_i are the coefs. Above is my understanding so far about relations brought by boundary operator, and now I am asked to show that sum of these coefs is 0, namely: $$∑_ia_i=0$$, I don't even see why this is even true. How should I approach this? Thanks for any suggestions.

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  • $\begingroup$ What is $U$ here? $\endgroup$ – 5xum Oct 22 '15 at 21:46
  • $\begingroup$ Edited. It's open set in R^n $\endgroup$ – jfcjohn Oct 22 '15 at 21:49
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    $\begingroup$ It's been a long time since I've dealt with this, but I'm pretty sure that the it results from the fact that $\partial^2 = 0$. $\endgroup$ – Paul Sinclair Oct 23 '15 at 0:01

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