[math-fun] The Wittless Duck
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 ... Fred Lunnon
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 ...
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 ...
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 30/01/2016 16:57, Fred Lunnon 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.
The definition at http://math.uga.edu/~pete/quadraticforms.pdf (which was the first hit I got for <<"witt index" "quadratic form">>) seems to imply a different value for the Witt index. Theorem 7.6 there says, I'm understanding it right, that a quadratic space has an orthogonal decomposition as R+D+H where - R is the "radical" and consists of vectors orthogonal to everything - D is anisotropic, i.e. nothing in it has q(v,v)=0 - H is a sum of "hyperbolic planes" (spanned by vectors v,w with q(v,v)=1, q(w,w)=-1, q(v,w)=0) and the Witt index is the number of hyperbolic planes. So, e.g., if the space is R^n and the quadratic form is the usual Euclidean inner product then in your notation we have p=n, q=0, r=0 so "your" Witt index is n; but the decomposition here is 0 + whole_space + 0 and the Witt index is 0. What am I missing? (Almost certainly something obvious. I don't know anything about this stuff.) -- g
What we're missing is that I mistyped the formula, which I have now recovered again from my notes that clearly state << The dimension of an isotropic sub-subspace cannot exceed the "Witt index" min(p,q) + r
(copied and pasted, this time). How on earth could I have misread that? Apologies to everybody! This question might be a good fit for some denizens in mathoverflow. I have come to the conclusion that a workable strategy to circumvent the trolls lurking under the drawbridge of that Tolkeinesque fortress is to surf the site for a related query, then search for email addresses of useful respondents (not usually very difficult) in order to approach them individually. Or maybe I'll just have to wait until I come across that theorem again by accident ... Fred Lunnon On 1/31/16, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 30/01/2016 16:57, Fred Lunnon 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.
The definition at http://math.uga.edu/~pete/quadraticforms.pdf (which was the first hit I got for <<"witt index" "quadratic form">>) seems to imply a different value for the Witt index.
Theorem 7.6 there says, I'm understanding it right, that a quadratic space has an orthogonal decomposition as R+D+H where - R is the "radical" and consists of vectors orthogonal to everything - D is anisotropic, i.e. nothing in it has q(v,v)=0 - H is a sum of "hyperbolic planes" (spanned by vectors v,w with q(v,v)=1, q(w,w)=-1, q(v,w)=0) and the Witt index is the number of hyperbolic planes. So, e.g., if the space is R^n and the quadratic form is the usual Euclidean inner product then in your notation we have p=n, q=0, r=0 so "your" Witt index is n; but the decomposition here is 0 + whole_space + 0 and the Witt index is 0.
What am I missing?
(Almost certainly something obvious. I don't know anything about this stuff.)
-- g
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Groan ... trying again: << The dimension of an isotropic subspace cannot exceed the "Witt index" min(p,q) + r
[ This theorem is becomes almost obvious with familiarity, unlike my typos. A basis for the maximal isotropic subspace ("kernel") comprises all the null generators z_i where (z_i)^2 = 0 , together with all pairs x_i - y_i where (x_i)^2 = +1 and (y_i)^2 = -1 ; obviously the latter number min(p,q) . Linear independence prevents any generator from being re-used. ] Fred Lunnon On 2/4/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
What we're missing is that I mistyped the formula, which I have now recovered again from my notes that clearly state
<< The dimension of an isotropic sub-subspace cannot exceed the "Witt index" min(p,q) + r
(copied and pasted, this time). How on earth could I have misread that? Apologies to everybody!
This question might be a good fit for some denizens in mathoverflow. I have come to the conclusion that a workable strategy to circumvent the trolls lurking under the drawbridge of that Tolkeinesque fortress is to surf the site for a related query, then search for email addresses of useful respondents (not usually very difficult) in order to approach them individually.
Or maybe I'll just have to wait until I come across that theorem again by accident ...
Fred Lunnon
On 1/31/16, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 30/01/2016 16:57, Fred Lunnon 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.
The definition at http://math.uga.edu/~pete/quadraticforms.pdf (which was the first hit I got for <<"witt index" "quadratic form">>) seems to imply a different value for the Witt index.
Theorem 7.6 there says, I'm understanding it right, that a quadratic space has an orthogonal decomposition as R+D+H where - R is the "radical" and consists of vectors orthogonal to everything - D is anisotropic, i.e. nothing in it has q(v,v)=0 - H is a sum of "hyperbolic planes" (spanned by vectors v,w with q(v,v)=1, q(w,w)=-1, q(v,w)=0) and the Witt index is the number of hyperbolic planes. So, e.g., if the space is R^n and the quadratic form is the usual Euclidean inner product then in your notation we have p=n, q=0, r=0 so "your" Witt index is n; but the decomposition here is 0 + whole_space + 0 and the Witt index is 0.
What am I missing?
(Almost certainly something obvious. I don't know anything about this stuff.)
-- g
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participants (3)
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Dan Asimov -
Fred Lunnon -
Gareth McCaughan