4-dimensional topology and physics

[BruceG]

It is well known to mathematicians that the study of topology in 4-dimensions is more difficult than in higher dimensions due to a "lack of freedom".

See for example http://hypercomplex.xpsweb.com/articles/146/en/pdf/01-09-e.pdf"

Further, as mentioned in this article, some of the developments of 4-dimensional topology has been inspired by developments in mathematical physics: in particular Freedman and Donaldson used Yang-Mills theory to study 4-dimensional manifolds.

My quesion then, is whether any of these ideas have fed back into professional research (string theory, M-theory or otherwise) to justify the existence of 4-dimensional spacetime (after compactification).

If this has been addressed on this, or another forum, please could you refer me.

 

[marcus]

Some paper(s) by Hendryk Pfeiffer (Cambridge math?)

I found it:
http://arxiv.org/abs/gr-qc/0404088
Quantum general relativity and the classification of smooth manifolds
Hendryk Pfeiffer
(Submitted on 21 Apr 2004 (v1), last revised 17 May 2004 (this version, v2))
The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory.
Comments: 41 pages

I see that Hendryk is now at UBC-Vancouver.
http://www.math.ubc.ca/~pfeiffer/
In 2004 when the paper was written he was at Cambridge DAMTP

Hopefully other people here will think of other papers and provide you with links. Some of Hendryk's work connects with LQG/Spinfoam models in various ways, other of his work seems to connect with John Baez' interests (categories, higher gauge theory, algebra...). Looks like an interesting mix of research.

 

[MTd2]

I guess you are looking for Torsten and Helge's paper about exotic smoothness. Check this out here:

https://www.physicsforums.com/showthread.php?t=412582

 

[BruceG]

Thanks for replies.
It seems that "exotic smoothness" is the starting point for me to look at.

Maybe I should start with this book:
Exotic Smoothness and Physics
Torston Asselmeyer-Maluga, Carl Brans

http://books.google.co.uk/books?id=...esnum=10&ved=0CFoQ6AEwCQ#v=onepage&q&f=false"