Donaldson's theorem
Encyclopedia
In mathematics
, Donaldson's theorem states that a definite intersection form
of a simply connected smooth manifold of dimension
4 is diagonalisable
. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix
(negative identity matrix). It was proved by Simon Donaldson
.
Michael Freedman
had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold. Combining this result with the Serre Classification Theorem and Donaldson's theorem, several interesting results can be seen:
1) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold
with no differentiable structure (so cannot be smoothed).
2) Two smooth simply-connected 4-manifolds are homeomorphic, if and only if, their intersection forms have the same rank
, signature, and parity.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Donaldson's theorem states that a definite intersection form
Intersection form
Intersection form may refer to:*Intersection theory *intersection form...
of a simply connected smooth manifold of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
4 is diagonalisable
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
(negative identity matrix). It was proved by Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
.
Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold. Combining this result with the Serre Classification Theorem and Donaldson's theorem, several interesting results can be seen:
1) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
with no differentiable structure (so cannot be smoothed).
2) Two smooth simply-connected 4-manifolds are homeomorphic, if and only if, their intersection forms have the same rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
, signature, and parity.