Thanks for this! A great way to start the year :)

You discuss two things close to my heart: algebraic topology as a driver of improvements in computational physics *and* the “cochain problem” involving the failure of cup product to be graded commutative “on the nose”. Of course, my interest in both stem from the fact I think the two are related.

I didn’t quite understand the bit about Eilenberg-Zilber map (EZ) and Alexander-Whitney map (AW). Are these related to the de Rham map (R) and the Whitney map (W)?

I wish I knew the correct “maths” way to express this, but given a manifold $M$ and differential forms $\Omega(M)$ on $M$, we can construct a simplicial complex $S$ with cochains $C^*(S)$. The de Rham map $R$ takes forms on $M$ and turns them into cochains on $S$

$R:\Omega(M)\to C^*(S).$

The Whitney map (W) takes cochains on $S$ and turns them into (Whitney) forms on $M$

$W:C^*(S)\to\Omega(M).$

We have

$R\circ W = Id_{C^*(S)}$

and

$W\circ R \sim Id_{\Omega(M)},$

which sounds similar to EZ and AW.

As you pointed out, the wedge product in $\Omega(M)$ is graded commutative and the cup product in $C^*(S)$ is not graded commutative “on the nose” so all these maps do not quite fit together perfectly, e.g.

$W(a\smile b) \ne W(a)\wedge W(b)$

and

$R(\alpha\wedge\beta) \ne R(\alpha)\smile R(\beta).$

Some people have proposed a modified cup product for computational physics via

$a\tilde\smile b := R(W(a)\wedge W(b)).$

I don’t think this helps much with the “cochain problem”. In particular, this modified cup product is not even associative!

I have proposed an alternative, which at first will seem radical but I actually think might help with things like the “cochain problem”. That is to introduce a modified wedge product

$\alpha\tilde\wedge\beta := W(R(\alpha)\smile R(\beta)).$

How dare we mess with the continuum :)

This modified wedge product has the (uber) nice property that it *is* an algebra homomorphism “on the nose”, i.e.

$R(\alpha\tilde\wedge\beta) = R(\alpha)\smile R(\beta).$

The undesirable property of this modified wedge product is that it will depend on $S$, but that dependence disappears when you pass to cohomology, homotopy, etc, which is what the mathematicians really care about.

My gut tells me that having a true algebra morphism like this will give you true functors and the category theoretic analysis will be much cleaner. But that is just a hunch.

Also note that in a suitable limit of refinements of $S$, i.e. a kind of “continuum limit” we have

$\alpha\wedge\beta = \lim_{continuum} \alpha\tilde\wedge\beta.$

## Re: This Week’s Finds in Mathematical Physics (Week 288)

John wrote:

for us oldsters is the adjoint pair: realization denoted | | and Sing

This preceded Quillen but I would guess that it inspired him