For a 4-manifold , its Donaldson invariants are an integer and maps (into half-integers), which combine into the Donaldson polynomial:[1][2]
Peter Kronheimer and Tomasz Mrowka introduced a condition known as Kronheimer–Mrowka simple type (KM 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 (KM basic classes), as well as rational numbers , called Kronheimer–Mrowka coefficients (KM coefficients), so that:
for all . Furthermore for all Kronheimer–Mrowka basic classes.[3][4][5]
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 (hence the first Chern class of spinc structures with a non-vanishing Seiberg–Witten invariant) and their Kronheimer–Mrowka coefficients are up to a topological factor exactly their Seiberg–Witten invariants. More concretely, it claims that a compactconnectedsimply connected orientable smooth 4-manifold with odd is of Kronheimer–Mrowka simple type if and only if is of Seiberg–Witten simple type (meaning non-vanishing Seiberg–Witten invariants only come from zero-dimensional Seiberg–Witten moduli spaces by counting its points with a sign determined by their orientation). In this case the Donaldson polynomial is given by:[6]
![{\displaystyle \gamma _{d}(M)\colon H_{2}(M,\mathbb {Z} )\rightarrow \mathbb {Z} [1/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/951f2952e12550d09ef935c2a3ca93de9eb5b557)
