Kronheimer–Mrowka basic class
In mathematics, the Kronheimer–Mrowka basic classes are elements of the second cohomology of a simply connected, smooth 4-manifold of simple type that determine its Donaldson polynomials. They were introduced by.
Description
For a 4-manifold, its Donaldson invariants are an integer and maps, which combine into the Donaldson polynomial:Peter Kronheimer and Tomasz Mrowka introduced a condition known as Kronheimer–Mrowka simple type, which is sufficient to obtain the separate Donaldson invariants from their common Donaldson polynomial. For a KM-simple manifold there are cohomology classes, called Kronheimer–Mrowka basic classes, as well as rational numbers, called Kronheimer–Mrowka coefficients, so that:
for all. Furthermore for all Kronheimer–Mrowka basic classes.
Although this reduction of the infinite sum of the Donaldson polynomial to a finite sum in early 1994 brought a significant simplification to Donaldson theory, it was overhauled just a few months later in late 1994 by the development of Seiberg–Witten theory. Edward Witten, presented in a lecture at MIT, used a purely physical argument to conjecture that Kronheimer–Mrowka basic classes are exactly the support of the Seiberg–Witten invariants and their Kronheimer–Mrowka coefficients are up to a topological factor exactly their Seiberg–Witten invariants. More concretely, it claims that a compact connected simply connected orientable smooth 4-manifold with odd is of Kronheimer–Mrowka simple type if and only if is of Seiberg–Witten simple type. In this case the Donaldson polynomial is given by: