[math-fun] Nonstandard smooth structure on R^4
The following will rely heavily on Alexandru Scorpan's book, The Wild World of 4-Manifolds. Part I. (There are a number of parts to this.) We'll talk about compact 4-manifolds. This will mean a compact topological space every point of which has a neighborhood homeomorphic to 4-space. (Or equivalently, to the open unit ball in R^4).* 0. We'll also assume all manifolds are simply connected (every loop can be shrunk to a point). 1. The integer homology group H_2[M] represents a way of thinking of [all the closed surfaces inside of M] as a group. It is always a finitely-generated free abelian group. Given a basis, we can view it as a direct sum Z^n of n copies of the integers Z. 2. Given any two homology classes u, v in H_2[M], there is an integer associated to them. First think of each as a surface in M. Then <u,v> := the number of intersection points of these surfaces. The points need to be counted with algebraic sign depending on the orientation of M. This may require perturbing them slightly to be in general position. It turns out to always be the same integer no matter how this is done. (As an analogy, think of the generic intersection of two closed curves in a closed surface — which also give an integer.) In fact, this mapping H_2[M] x H_2[M] -> Z is a symmetric bilinear form over the integers. Of special interest are the self-intersection numbers <u,u>. Restricting the bilinear form to self-intersections, it becomes a quadratic form Q: H_2[M] -> Z via Q(u) := <u,u>. This is called the INTERSECTION FORM of M. (Two such quadratic forms are "equivalent" if there is a Z-basis for each one such that the resulting symmetric matrices are the same.) FACT: Because of "Poincaré duality", Q is always UNIMODULAR, which means that det(Q) = ±1. Another property of quadratic forms is DEFINITENESS: If all values taken by Q(u) are postive, or else all values are negative, the form is respectively POSITIVE or NEGATIVE DEFINITE. Otherwise the form is INDEFINITE. 3. MAJOR RESULT: Theorem of Serre: ---------------- "Two indefinite unimodular integer quadratic forms are equivalent if and only if they have the same rank (the n here), signature, and parity." ----- The (symmetric integer) matrix of a quadratic form can always be diagonalized over the REALS. When this has been done, its SIGNATURE is the number of positive terms minus the number of negative terms on the diagonal. The parity is EVEN if the quadratic form takes only even values. Otherwise it is ODD. Unfortunately, any classification of *definite* quadratic forms over Z is currently out of reach. 4. MAJOR RESULT: Theorem of Michael Freedman, ca. 1981: ---------------- "The topological classification of simply connected compact 4-manifolds is as follows: There is at least one for each inequivalent quadratic form Q, having that form as its intersection form. "If Q is even: -------------- There is exactly one such manifold up to topological equivalence. "If Q is odd: ------------- There are two such manifolds, at least one of which cannot be smoothed." ----- Example: The simplest nonzero quadratic form has the matrix [1]. An example of this is the complex projective plane CP^2. This is not only smooth, but the nicest kind of complex analytic manifold: a Kähler manifold. But since [1] is an odd form, there must be another topological manifold with the same quadratic form that is not smoothable at all. -----END OF PART I----- --Dan ——Dan __________________________________________________________________ * To avoid pathologies, we also assume manifolds are Hausdorff and separable (have a countable dense subset).
Thanks Dan — I look forward to PART II. In the early 80’s a couple of us physicists asked Dan Freed (then Isadore Singer’s student) if the exotic R^4’s might have any relevance for physics and without going into details his answer was “no”. I’m now thinking I should have asked for those details. -Veit
On Apr 25, 2015, at 2:15 PM, Dan Asimov <asimov@msri.org> wrote:
The following will rely heavily on Alexandru Scorpan's book, The Wild World of 4-Manifolds.
Part I. (There are a number of parts to this.)
We'll talk about compact 4-manifolds.
This will mean a compact topological space every point of which has a neighborhood homeomorphic to 4-space. (Or equivalently, to the open unit ball in R^4).*
0. We'll also assume all manifolds are simply connected (every loop can be shrunk to a point).
1. The integer homology group H_2[M] represents a way of thinking of [all the closed surfaces inside of M] as a group.
It is always a finitely-generated free abelian group. Given a basis, we can view it as a direct sum Z^n of n copies of the integers Z.
2. Given any two homology classes u, v in H_2[M], there is an integer associated to them. First think of each as a surface in M. Then
<u,v> := the number of intersection points of these surfaces.
The points need to be counted with algebraic sign depending on the orientation of M. This may require perturbing them slightly to be in general position. It turns out to always be the same integer no matter how this is done. (As an analogy, think of the generic intersection of two closed curves in a closed surface — which also give an integer.)
In fact, this mapping H_2[M] x H_2[M] -> Z is a symmetric bilinear form over the integers. Of special interest are the self-intersection numbers <u,u>. Restricting the bilinear form to self-intersections, it becomes a quadratic form
Q: H_2[M] -> Z
via
Q(u) := <u,u>.
This is called the INTERSECTION FORM of M.
(Two such quadratic forms are "equivalent" if there is a Z-basis for each one such that the resulting symmetric matrices are the same.)
FACT: Because of "Poincaré duality", Q is always UNIMODULAR, which means that det(Q) = ±1.
Another property of quadratic forms is DEFINITENESS: If all values taken by Q(u) are postive, or else all values are negative, the form is respectively POSITIVE or NEGATIVE DEFINITE. Otherwise the form is INDEFINITE.
3. MAJOR RESULT: Theorem of Serre: ----------------
"Two indefinite unimodular integer quadratic forms are equivalent if and only if they have the same rank (the n here), signature, and parity." -----
The (symmetric integer) matrix of a quadratic form can always be diagonalized over the REALS. When this has been done, its SIGNATURE is the number of positive terms minus the number of negative terms on the diagonal.
The parity is EVEN if the quadratic form takes only even values. Otherwise it is ODD.
Unfortunately, any classification of *definite* quadratic forms over Z is currently out of reach.
4. MAJOR RESULT: Theorem of Michael Freedman, ca. 1981: ----------------
"The topological classification of simply connected compact 4-manifolds is as follows: There is at least one for each inequivalent quadratic form Q, having that form as its intersection form.
"If Q is even: --------------
There is exactly one such manifold up to topological equivalence.
"If Q is odd: -------------
There are two such manifolds, at least one of which cannot be smoothed." -----
Example: The simplest nonzero quadratic form has the matrix [1]. An example of this is the complex projective plane CP^2. This is not only smooth, but the nicest kind of complex analytic manifold: a Kähler manifold.
But since [1] is an odd form, there must be another topological manifold with the same quadratic form that is not smoothable at all.
-----END OF PART I-----
--Dan
——Dan __________________________________________________________________ * To avoid pathologies, we also assume manifolds are Hausdorff and separable (have a countable dense subset). _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Apr 27, 2015, at 7:51 AM, Veit Elser <ve10@cornell.edu> wrote:
Thanks Dan — I look forward to PART II.
Coming soon.
In the early 80’s a couple of us physicists asked Dan Freed (then Isadore Singer’s student) if the exotic R^4’s might have any relevance for physics and without going into details his answer was “no”. I’m now thinking I should have asked for those details.
Didn't Dan Freed write or coauthor a book about 4-manifolds? (Btw, as an undergrad, Singer was my favorite prof — I had him for three courses.) --Dan
participants (2)
-
Dan Asimov -
Veit Elser