From: gt0109e@prism.gatech.edu (Frank Dodd Smith)
Subject: exotic S7
To: rusin@math.niu.edu
Date: Thu, 27 Jul 1995 21:29:33 -0400 (EDT)

I have had trouble today trying to download your 

ftp://ftp.math.niu.edu/pub/papers/Rusin/math.topics/exotic.s7 

from the WWW using Netscape for Mac.  

I can get other papers just fine, but that one does not work.  

The reason for my interest in it is to try to find out 
an explicit description of the 7-dim fibre space M7 of 
the fibration of the 23-sphere S23 
with base manifold Cayley projective plane OP2 
      
        M7 --> S23 --> OP2  

I saw something on sci-math-research years ago that 
said M7 could not be the 7-sphere S7, but it did not 
say what M7 might be. 

I also saw a paper of Adams (Topology 1 (1961) 67-72) 
that said a 7-sphere bundle over the 15-sphere could  
possibly be an H-space, but that (at the time) Adams 
did not know whether or not it was.  

Any help you could give me in finding an explicit 
construction of M7 would be much appreciated.  

Tony Smith 27 Jul 95
e-mail:  gt0109e@prism.gatech.edu
WWW:  http://www.gatech.edu/tsmith/home.html 
      http://galaxy.cau.auc.edu/tsmith/TShome.html

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

Date: Thu, 27 Jul 95 23:23:23 CDT
From: rusin (Dave Rusin)
To: gt0109e@prism.gatech.edu
Subject: Re:  exotic S7

Sorry, had the file permissions set wrong. Should work now.

I'm not sure that "paper" will help you. The issue addressed in it
is the classification of diffeomorphism classes of topological spheres;
in the case of S^7 there are 28 such, distinguishable by a single invariant.
Is that your goal? Or are you trying to establish that  M7  is homeomorphic
to a sphere? (That part sounds straightforward, given the two
fibrations involving  OP2  and the Poincare conjecture.)

Milnor's paper (1957? Annals?) classifying the diffeomorphism classes
is really quite short and easy. The description of these spheres as
hypersurfaces near an algebraic singularity is I think due to Brieskorn
around the early 70s; the basic ideas in that business are also due
to Milnor and may be found in his "Singular Points of Complex Hypersurfaces"
(Princeton U Press). Milnor discusses there some fibrations which may be
similar to the ones you are studying.

Let me know if there's anything else I can do for you.

dave

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

From: Frank Smith <tsmith@mordred.gatech.edu>
Subject: S7 and M7
To: rusin@math.niu.edu
Date: Fri, 28 Jul 1995 04:45:21 -0400 (EDT)

Thanks for the information.  The ftp works fine now.  
It is a nice "paper".  
  
I have Milnor's book on complex hypersurfaces, and it is good. 

The purpose of my question is to try to understand 
octonion structures related to the 24-dim Leech lattice.  

S7 is not a Lie group, and is not generated by a Lie algebra, 
but there is a Malcev algebra structure (based on torsion due 
to non-associativity of octonions).  

Martin Cederwall 
(WWW URL  http://xxx.lanl.gov/abs/hep-th/9309030)
characterized the Malcev algebra in terms of a 
generalization of the octonion product that he called 
an X-product, a product that varies with the points X of S7. 

Then Geoffrey Dixon 
(WWW URL  http://xxx.lanl.gov/abs/hep-th/9503053)
described an XY-product for octonions, 
and related the X-product to the 7 different discrete 
E8 lattices (and therefore to the 7 diffferent sets 
of 240 points that are nearest neighbors to the origin)
(WWW URL  http://xxx.lanl.gov/abs/hep-th/9411063). 

He then 
related the XY-product to the ways 
the discrete S7 in the E8 lattice could be used to 
fibre the discrete S15 in the 16-dim Barnes-Wall lattice /\16, 
and related that to a discrete version of the 
last Hopf map    S7  ->  S15  ->  S8 = OP1. 
(WWW URL  http://xxx.lanl.gov/abs/hep-th/9501007)

Now he is working on the ways the discrete S7 in the E8 lattice 
could be used to fibre the discrete S23 in the 24-dim Leech lattice /\24.
(WWW URL  http://xxx.lanl.gov/abs/hep-th/9504040)
(WWW URL  http://xxx.lanl.gov/abs/hep-th/9506080)

A question is, if M7 is not diffeomorphic to S7 
(as you say, they are probably homeomorphic) 
then is the differentiable structure of M7 not one of the 28 of S7 ?  
If it differs, how does it differ? 

Intuitively to me (and my intuition is often fallible), 
it seems that 
the X-product is related to 7 of the 28 differentiable structures
(one of the two S7s in 28-dim Spin(8)) 
and 
the XY-product to 7 more of the 28 differentiable structures
(the other S7 in Spin(8)) 
but 
I do not know of a generalized product that is related 
to the remaining 14 differentiable structures
(the remaining G2 in Spin(8), i.e. 
the G2 in the fibration S7 -> Spin(7) -> G2). 


Thanks again for your help. 

Tony 28 Jul 95
e-mail:  gt0109e@prism.gatech.edu
WWW:  http://www.gatech.edu/tsmith/home.html 
      http://galaxy.cau.auc.edu/tsmith/TShome.html