From: Alan Pengelly <apengell@srd.bt.co.uk>
Date: Wed, 2 Aug 95 09:26:31 BST
To: rusin@math.niu.edu
Subject: Re: 7-SPHERE

> In article <3vkt38$5h6@pheidippides.axion.bt.co.uk> you write:
> >Hi
> >
> >Could someone please point me to a reference which goes into
> >some depth on all the properties and characteristics of the 7-sphere.
> 
> Why  7  in particular? Just about everything you need to know about
> the spheres is true in all dimensions, or at least those of dimensions
> 4 or more. There are only a couple of features I know which distinguish
> S^7:
> 

I've actually been reading up some theoretical cosmology where, in the context
of the `superforce' etc, the 7-sphere is mentioned as having some exotic
properties. Coupled with 4-d space-time, we have an 11-d manifold.

> 1) S^7 is parallelizable (and  7  is the largest such exponent), a
> consequence of the existence of the 8-dimensional division algebra
> called the Cayley algebra or Octonions. See e.g. Milnor's book
> Characteristic classes (princeton Univ press)
> 
> 2) S^7 admits several inequivalent differentiable structures (and
> 7  is the smallest such exponent). Below is an old post I made relating to 
> this.

Is this the property that there are surfaces which are homeomorphic
but not diffeomorphic?
 
> I think Milnor writes really well, which is why I recommend his work
> in general.
> 
> dave
> 
Thanks for your help.

regards

Alan 

==============================================================================

Date: Wed, 2 Aug 95 10:54:25 CDT
From: rusin (Dave Rusin)
To: apengell@srd.bt.co.uk
Subject: Re: 7-SPHERE

>I've actually been reading up some theoretical cosmology where, in the context
>of the `superforce' etc, the 7-sphere is mentioned as having some exotic
>properties. Coupled with 4-d space-time, we have an 11-d manifold.

Cool. Knowing nothing about cosmology past Physics 101 I might guess that
the 'exotic' properties are related to the multiplicative structure of
S^7 (it's an "H-space", meaning that there's a product which is not
associative but close to it); this would give extra structure to the
analysis -- you know, like the "8-fold way" in particle physics is the
result of group symmetries.

>Is this the property that there are surfaces which are homeomorphic
>but not diffeomorphic?

Precisely. (Don't call them surfaces; that suggests being 2-dimensional.
They are manifolds.) I would be surprised if this turns out to be relevant
to the applications you described above. On the expectation, then, that
what you need to do is bone up on the Cayley algebra instead, I'll give
you a URL to another post I made which discusses this and other product
structures. It's kind of long and not a little off-topic for you, but you
might want to scan for relevant parts. It's actually a couple of files:
	ftp://math.niu.edu/pub/papers/Rusin/products/*.*
(you can also get it by gopher from that address, or with a Web browser
connect to http://www.math.niu.edu/~rusin  and look around.)

dave