Kronheimer–Mrowka basic class

For a 4-manifold ${\displaystyle M}$, its Donaldson invariants are an integer ${\displaystyle \gamma _{0}(M)\in \mathbb {Z} }$ and maps ${\displaystyle \gamma _{d}(M)\colon H_{2}(M,\mathbb {Z} )\rightarrow \mathbb {Z} [1/2]}$ (into half-integers), which combine into the Donaldson polynomial:^[1][2]

${\displaystyle {\mathcal {D}}_{M}\colon H_{2}(M,\mathbb {Z} )\rightarrow \mathbb {R} ,{\mathcal {D}}_{M}(x)=\sum _{d=0}^{\infty }{\frac {\gamma _{d}(M)(x)}{d!}}.}$

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 ${\displaystyle M}$ there are cohomology classes ${\displaystyle K_{1},\ldots ,K_{s}\in H^{2}(M,\mathbb {Z} )}$, called Kronheimer–Mrowka basic classes (KM basic classes), as well as rational numbers ${\displaystyle a_{1},\ldots ,a_{s}\in \mathbb {Q} }$, called Kronheimer–Mrowka coefficients (KM coefficients), so that:

${\displaystyle {\mathcal {D}}_{M}(x)=\exp(Q_{M}(x,x)/2)\sum _{r=1}^{s}a_{r}\exp(K_{r}(x))}$

for all ${\displaystyle x\in H_{2}(M,\mathbb {Z} )}$. Furthermore ${\displaystyle w_{2}(M)=K_{r}\operatorname {mod} 2\in H^{2}(M,\mathbb {Z} _{2})}$ 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 ${\displaystyle \operatorname {SW} \colon \operatorname {Spin} ^{\mathrm {c} }(M)\rightarrow \mathbb {Z} }$ (hence the first Chern class ${\displaystyle c_{1}\colon \operatorname {Spin} ^{\mathrm {c} }(M)\rightarrow H^{2}(M,\mathbb {Z} )}$ of spin^c 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 compact connected simply connected orientable smooth 4-manifold ${\displaystyle M}$ with ${\displaystyle b_{2}^{+}(M)\geq 2}$ 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]

${\displaystyle {\mathcal {D}}_{M}(x)=\exp(Q_{M}(x,x)/2)\sum _{{\mathfrak {s}}\in \operatorname {Spin} ^{\mathrm {c} }(M),\dim {\mathcal {M}}_{\mathfrak {s}}^{\mathrm {SW} }=0}2^{2+{\frac {1}{4}}(7\chi (M)+11\sigma (M))}\operatorname {SW} (M,{\mathfrak {s}})\exp(c_{1}({\mathfrak {s}})(x)).}$

• Kronheimer, Peter B.; Mrowka, Tomasz S. (1994), "Recurrence relations and asymptotics for four-manifold invariants", Bulletin of the American Mathematical Society, New Series, 30 (2): 215–221, arXiv:math/9404232, doi:10.1090/S0273-0979-1994-00492-6, ISSN 0002-9904, MR 1246469
• Kronheimer, Peter B.; Mrowka, Tomasz S. (1995), "Embedded surfaces and the structure of Donaldson's polynomial invariants", Journal of Differential Geometry, 41 (3): 573–734, doi:10.4310/jdg/1214456482, ISSN 0022-040X, MR 1338483
• Naber, Gregory L. (2011). Topology, Geometry and Gauge Fields. Applied Mathematical Sciences. Vol. 141 (Second ed.). Springer. doi:10.1007/978-1-4419-7895-0. ISBN 978-1-4419-7894-3. ISSN 0066-5452.