Demystifying math: How someone you could have come up with singular homology?

2017-12-14 05:23:06

I am having trouble with finding the best intuition for homology as sometimes I'm able to find for other subjects which is how someone could have come up with it.

Reading the history of homology didn't satisfy me, I was wondering if someone could help me with this. (The other answers on math exchange aren't really what I'm looking for, they somehow already assume we know what the homology groups are).

I would love an answer that is to homology as those short ones (mostly because I can't think of a long one) illustrate the kind of motivation I like best:

Homotopy: It is clearly interesting to see the maps from $S^1$ to our space look like. After some thinking, we realize we can impose a group structure if we fix the base point, but do we really want to? Umm we kind of get stuck otherwise, so let's fix a basepoint to make things simpler, in fact we also see (details left) after some thinking that it doesn't matter which one if the space is connected!

Ok let's take a si

  • Answering how the boundary operator is a natural thing to consider in homology: Homology (singular, simplicial, and to a lesser extent, cellular) basically asks whether one can, given an $n$-sphere in a space, fill in the sphere to make an $(n+1)$-ball. It's a very natural intrinsic way to look for "holes" in the space.

    Rephrasing, we're asking whether something looking like the boundary of an $(n+1)$-ball actually is the boundary of such a ball. How do we tell that something looks like the boundary of a ball? That it is $n$-dimensional, doesn't have boundary itself, and is orientable (which may be taken care of by giving a sign to the boundary operator) is a good start.

    That turns out to be more or less enough as well. A thing that fulfills these criteria might not look like the boundary of a sphere (it could be a torus, for instance). But whatever it looks like the boundary of, that thing can be glued together from balls. And because of the sign convention on the boundary operato

    2017-12-14 05:49:55
  • I know this answer will diverge somewhat from the topological flavor of the original question - but I thought I'd share some thoughts on how somebody "could have" come up with homological algebra from an algebraic point of view.

    So, suppose you have a short exact sequence $0 \to A \to B \to C \to 0$ of, say, $R$-modules; then you have a morphism $f : U \to C$, and you want to determine whether there is a lifting of this morphism to a morphism $\tilde f : U \to B$. (If you like, you can suppose that $R$ is Noetherian, and each module is finitely generated.) Once you realize this is not always possible, but it is possible if $U$ is a free $R$-module, then you might start to investigate how to determine from $U$ what sorts of "obstructions" there are to constructing a lifting map.

    So, you would start off by taking some set of generators of $U$, and then try to map each generator $x$ to some preimage of $f(x)$ in $B$. Now, the reason this doesn't always work is that there are also r

    2017-12-14 05:51:00