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By Kaczynski , Mischaikow , Mrozek

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In other words, what kind of constraint on f# will guarentee that cycles which are boundaries go to boundaries? To answer this lets repeat what we have said. c is a boundary so we can write c = @b for some chain b. Thus f#(c) = f#(@b). But we want f#(c) to be the boundary of some chain. What chain? The only one we have at our disposal is b, so the easiest constraint is to ask that f#(c) = @f# (b).

If you feel things are spinning out of control - don't worry, be happy! Admittedly, there are a lot of loose ends that we need to tie up and we will begin to do so in the next chapter. The process of developing new mathematics typically involves developing new intuitions and nding new patterns - in this case we have the advantage of knowing that it will all work out in the end. For now lets just enjoy trying to match topology and algebra. In fact, lets do it again. Recall that earlier we asked the question what should be use for scalars?

These problems have their origins in topology (not surprising), computer graphics, dynamical systems, parallel computing, and numerics. Obviously for such a broad set of issues a single chapter cannot do any of the topics justice. They are included solely for the purpose of motivating the formidable algebraic machinery we are about to start developing. This chapter is meant to be enjoyed in the sense of an entertaining story. Don't sweat the details - try to get a feeling for the big picture. We will return to these topics throughout the rest of this book.

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