The n-Category Café

John said:

Maybe we should vote on some result to call the Fundamental Theorem of Topology — but with one condition, that its only known proof is algebraic.

Does the use of calculus for a topological result count as an impurity?

I’m thinking about the Riemann mapping theorem, you could try to prove the topological content of it using topological methods only. But I only know about proofs (both constructive and not constructive) using complex calculus to prove the stronger statement of the original theorem, therefore using the multiplicative structure of the complex numbers in a critical way, which is in turn irrelevant for the topological part.

Wikipedia states:

The Riemann mapping theorem is the easiest way to prove that any two simply connected domains in the plane are homeomorphic. Even though the class of continuous functions is vastly larger than that of conformal maps, it is not easy to construct a one-to-one function onto the disk knowing only that the domain is simply connected.

…which seems to indicate that there is a - constructive(?) - proof of the topological part of the theorem which does not use complex analysis, which I don’t know…

Thanks for blowing the trumpet for me.
But the James construction goes the other way -
builds up to an approximation of based loops on a suspension space.

A topological monoid M is in particular an A_\infty-space
and either of the constuctions of the classiying space is filtered by projective spaces, e.g. BS^1= ..\superset CPn…\superset CP2 \superset CP1

Exactly! Not at all sure what kind of `coordinates’ there are for such projective spaces.

Perhaps more accessible place to read ab out htis would be

PDF Clipboard Journal Article
MR0270372 (42 #5261) Stasheff, James $H$H-spaces from a homotopy point of view. Lecture Notes in Mathematics, Vol. 161 Springer-Verlag,

David wrote:

So there would then be an obstruction to extending something 3-dimensional to something 4-dimensional, and so on? But this doesn’t manifest itself in ordinary projective spaces, since once the Desarguesian obstruction is removed, all dimensions are possible?

I seem to remember the story goes like this. If $G$ is a topological group you can define its nerve $B G$ as a topological space. When $G$ is the group of invertible elements in a topological field $F$ this is the infinite-dimensional projective space $F \mathbb{P}^\infty$. But you can also try to build up $F \mathbb{P}^\infty$ stage by stage:

$$F\mathbb{P}^1 \hookrightarrow F\mathbb{P}^2 \hookrightarrow F\mathbb{P}^3 \hookrightarrow \cdots$$

The points of $F\mathbb{P}^1$ are the 1-simplices in $B G$, the points of $F\mathbb{P}^2$ are the 2-simplices in $B G$, and so on. Or something like that: I feel I have some of the details slightly wrong.

To do the first few stages of this construction, we don’t need $F$ to be a full-fledged topological field. But it needs to have better and better properties as we go up. It goes something like this:

If $F$ is a space with an element $0$ you can define $F \mathbb{P}^1$. If it has a product where every nonzero element has an inverse you can define $F \mathbb{P}^2$. If the product is associative you can define $F \mathbb{P}^3$… and all the higher $F \mathbb{P}^n$’s!

That’s why there’s an octonionic projective plane but no higher octonionic projective spaces, and that’s why “once the Desarguesian obstruction is removed, all dimensions are possible.”

However, the story gets more involved if $F$, instead of having a product that’s associative on the nose, has a product that’s associative only up to homotopy. This is what Jim Stasheff worked out in his famous book.

For example: if $F$ is associative up to homotopy, we can define $\mathbb{F} P^3$ but not necessarily any higher $\mathbb{F} P^n$’s. If this homotopy obeys the pentagon identity, we can we can define $\mathbb{F} P^4$… and all the higher $\mathbb{F} P^n$’s!

I think you can guess the pattern from here on out.

Actually ‘fields’ are not required for most of this story to work. It’s only the multiplication we’re using here, not the addition. But it’s nice to think about fields (and their defective relatives, like the octonions) to make the connection between what Jim did and the old story of projective geometry. So, it might be nice to think about the projective geometry of ‘homotopy fields’, ‘$A_n$ fields’ and the like. Maybe people already have. I know they’ve thought a lot about ‘field spectra’, which usually go by some other name. All this is part of what some people call ‘brave new algebra’. But I don’t know if anyone has thought about this high-powered stuff by trying to generalize the old-fashioned axiomatic approach to projective geometry!