Hahn's 290 theorem --- "proved, I regret to say, using a computer", as Christopher Hooley would surely have observed --- is indeed new to me. However, as I failed to make clear earlier, the spaces in which I am interested are distressingly conventional, with real, complex, or (at a pinch) p-adic base rings. Which would also be true of most of the audience for "Advances in Applied Clifford Algebras", I should have thought --- making this a somewhat unexpected venue for such number theoretic material! WFL On 1/30/16, Dan Asimov <asimov@msri.org> wrote:
Don't know if this contains what you're thinking of, Fred, but it certainly is a fascinating survey of quadratic forms over the integers, showing just how nontrivial this subject is:
https://math.nd.edu/assets/20630/hahntoulouse.pdf <https://math.nd.edu/assets/20630/hahntoulouse.pdf>.
The theory of quadratic forms over the integers is central to the theory of topological 4-manifolds, because of Michael Freedman's 1982 theorem (here cribbed from notes on the web):
----- Theorem 1. For each symmetric bilinear unimodular form Q over Z there exists a closed oriented simply-connected topological 4-manifold with Q as its intersection form.
If Q is even there is precisely one; if Q is odd there are precisely two, at least one of which is nonsmoothable. -----
—Dan
On Jan 30, 2016, at 8:57 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I have been struggling to recall a rather surprising theorem, encountered in passing while searching for something unrelated, concerning the structure of quadratic spaces for which the Witt index exceeds 2 . Not only can I not find the theorem; I can't even locate a definition of the Witt index --- that is |p - q| + r , where the (possibly degenerate) quadratic form signature involves p positive, q negative, r absent squares.
Can somebody out there please unscramble my brain (or my surfing technique) for me? Dribble, mutter ...
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