5
$\begingroup$

For instance:

  1. $ |A \cup B \cup C|=(|A|+|B|+|C|)-(|A \cap B| + |A \cap C| + |B \cap C|)+|A \cap B \cap C|$
  2. $\chi(X) = F- E + V = \sum_{i} (-1)^i \text{rank}(H_i(X)) =\sum_{i} (-1)^i \text{rank}(C_i(X))$
  3. Differential forms/Exterior algebra
  4. $\partial \sigma = \sum_{i=0}^n (-1)^i \sigma|[v_0, \dots,\hat{v_i}, \dots, v_n]$

Points (2), (3), and (4) all fall under "homology stuff" so I would expect there to be a consistent explanation of the signs in these cases. In particular, given my extremely rough understanding of differential forms and simplices being analogous I would expect there to be a precise explanation of the origin of the alternating-ness in these cases. I know you can attribute some of this to orientations and the symmetric group being present in both.

I see a nebulous relation between (1) and (3) in that differential forms and cardinality of finite sets are both measures of sorts. I might extend this to (2) after reading about the length of a potato, which in particular pointed out the relevance of the euler characteristic in making measurements.

There is also a relation between (1) and (2) in that the formula in (1) generalizes to: Given a finite set $X$ and finitely many subsets covering it, the cardinality of $X$ is the euler characteristic of the nerve of this cover. Note: The nerve is a simplicial set with one $n$-simplex for each $n$-fold intersection; I am defining the euler characteristic of a simplicial set to be the alternating sum $\sum_i (-1)^ic_i$ where $c_i$ is the number of non degenerate $n$-simplices.

This interpretation of (1) also fuels a connection to (4) because there are simplices in both.

The one general statement that I can make about all of the above examples, although this doesn't explain the alternating signs, is that the particular forms are often invariants under some sorts of transformations or representations. For instance (1) expresses the cardinality of a set in terms of any number of decompositions of that set into subsets; regarding (2), we know the euler characteristic is an invariant of homotopy; regarding (3), differential forms transform appropriately under diffeomorphisms; and regarding (4), the boundary is invariant under orientation preserving maps of the simplex.

What I'm looking for is a conceptual unified explanation of the alternating-ness in all of the above examples. Such an explanation would optimally expand on each of the points I made above. (I might also be looking for a better word that "alternating-ness", but thats another question.)

$\endgroup$
  • $\begingroup$ (1) is pure combinatorics. For (2),(3) and (4) I'd put the blame on orientability. $\endgroup$ – Ivo Terek Sep 5 '16 at 3:35
  • $\begingroup$ I'd be fascinated if (2) was due to orientability, but that seems far from the case. $\endgroup$ – Robin Allison Sep 25 '16 at 4:46
  • 1
    $\begingroup$ I also wonder about this. I think one reason is that these are manifestation of the inclusion-exclusion principle on combinatorics, the idea if over counting, then measuring the amount of over counting. (The first is literally that, the second is a homotopy invariant version of that. The fourth is a triangulation invariant sort of inclusion exclusion, when you want to count the faces of a simplex independently of how you subdivide it. The differential forms could be thought of as invariantly measuring stuff too, and Stokes theorem ...) I don't really know how to make this precise though. $\endgroup$ – Lorenzo Jul 24 '17 at 5:13
  • $\begingroup$ (2) comes directly from (1). $\endgroup$ – Peter Saveliev Apr 11 '18 at 1:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.