algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
Donaldson’s theorem states that the intersection form of a compact orientable 4-manifold is diagonalizable. It was proven by Simon Donaldson in 1983 in Donaldson 83 (with additionaly using simple-connectedness) and improved in 1987 in Donaldson 87 (without using simple-connectedness). It became one of the reasons for him getting the Fields Medal in 1986 due to the new method of using the moduli space of the anti-self-dual Yang-Mills equations (ASDYM equations) to study the base manifold, which became the ground of Donaldson theory.
Central consequences of Donaldson’s theorem are the existence of exotic smooth structures on euclidean space in four dimensions and the failure of the h-cobordism theorem in four dimensions.
Let be a compact orientable 4-manifold and be a principal -bundle. According to the Atiyah-Singer index theorem, the dimension of the moduli space of anti-self-dual Yang-Mills connections (ASDYM connections) is given by:
with the second Chern class , the fundamental class given by the orientation of , the Kronecker pairing (which is often obmitted), the first Betti number and the dimension of the positive definite subspace of with respect to the intersection form.
If considering a simply connected as in the original version, then and hence , which simplifies the formula. Consider a principal -bundle with , hence . Reducible connections modulo gauge, which are the singularities of , correspond to decompositions with a complex line bundle , which implies:
Let be the number of pairs with , then with equality iff is diagonalizable.
resembles the base manifold at infinity and the complex projective plane around the singularities. Gluing them in yields a compactification, which contains a cobordism between and disjoint . Since the signature is a coborism invariant, this implies:
The 4-sphere is a compact orientable 4-manifold. Although its intersection form is trivial, it gives insight into the concepts used in the proof and their relation to each other. The principal -bundle over with Chern class is the quaternionic Hopf fibration . It can abstractly be defined as the Hopf construction of the group structure on or more directly by the unit quaternions acting on both components of and the projection on the orbit space, which is the quaternionic projective space . The adjoint vector bundle is the quaternionic tautological line bundle (considered as complex plane bundle) again defined using the identification , in which points in correspond to one-dimensional linear subspaces of . One has and , hence and .According to the above formula, this yields , which for instantons corresponds to the four freedoms regarding position and the one freedom regarding size.
See also:
Last revised on June 28, 2024 at 11:14:05. See the history of this page for a list of all contributions to it.