Intuition behind alternating sum in boundary operator definition?

Question

Boundary of simplex is formal alternating sum of its faces.

It is clear for me, that with ordinary sum main principle $dda = 0$ will not hold for ordinary sum. For every n+2-simplex $a$, there will be two copies of every it's n-subsimplex, and if we use alternating sum they all will kill each other.

But I can not understand, what all this signs really mean.

Is there reasons why boundary operator must be defined like this? Can the definition be derived from simpler properties, as for Euler charachteristic?

Answer 1

To understand this intuition you should read the history of singular theory, and the relation of homology theory to integration theory: see for example S. Lefschetz's article in I.M James (editor) "History of Topology". It was Poincare who started homology theory proper with the notion of formal sums of oriented simplices. The notion of orientation was essential in integration on higher dimensional domains, i.e. subsets of $\mathbb R^n$, and the notion of "formal sum" came from the convenient notation $$\int _C f dz + \int _D f dz = \int _{C + D} f dz . $$ However it was also convenient to allow domains as "singular" simplices, i.e. as maps $s:\Delta^n \to \mathbb R^m$. This then led to difficulties with "degenerate maps", which "squash" a simplex, leading to chain groups with elements of order $2$.

Finally Eilenberg in his 1945 paper on "Singular Homology" avoided this difficulty by using ordered simplices, and this led to our current simplicial theory. For more on oriented theory, see this paper by Barr.

For more discussion on "anomalies in algebraic topology" see this presentation.

Answer 2

I see it this way:

• When you define an $n$-simplex using an (ordered) tuple of $n+1$ points, you implicitly define an orientation on the set of n-simplices with the same vertices : two such n-simplices have the same orientation if they are related through an even permutation.
• Consider the $(n-1)$-simplices you can obtain by removing one of the previous points. You may want to inherit the orientation defined previously to define a "compatible" orientation on those $(n-1)$-simplices.
• (*) One way to do this is to state : $[v_0, ..., v_{j-1}, v_{j-1}, ..., v_n]$ is positively oriented iff $[v_j, v_0, ..., v_{j-1}, v_{j-1}, ..., v_n]$ is positively oriented — I'm using the notations of https://en.wikipedia.org/wiki/Simplex#Algebraic_topology.
• If you take that as a definition for an induced orientation, you now see that the orientation of $[v_0, ..., v_{j-1}, v_{j-1}, ..., v_n]$ is related to the one of $[v_0, ..., v_{j-1}, v_j, v_{j-1}, ..., v_n]$ by a factor $(-1)^j$ (corresponding to $j$ transpositions), which justifies the formula for the boundary.

Now a question is why (*) should be a good way to define an induced orientation ? It actually gives a definition of an induced orientation compatible with the one for a manifold with boundary.