Kronheimer–Mrowka basic class

In mathematics, the Kronheimer–Mrowka basic classes are elements of the second cohomology $H^2(M;\mathbb{\mathbb{Z}})$ of a simply connected, smooth 4-manifold $M$ of simple type that determine its Donaldson polynomials. They were introduced by Peter B. Kronheimer and Tomasz S. Mrowka (1994, 1995).

Description

For a 4-manifold $M$, its Donaldson invariants are an integer ${\gamma }_0(M)\in \mathbb{\mathbb{Z}}$ and maps ${\gamma }_d(M):H_2(M,\mathbb{\mathbb{Z}})\to \mathbb{\mathbb{Z}}[1/2]$ (into half-integers), which combine into the Donaldson polynomial:

${{𝒟}}_M:H_2(M,\mathbb{\mathbb{Z}})\to \mathbb{ℝ},{{𝒟}}_M(x)=\sum _{d=0}^{\mathrm{\infty }}\frac{{\gamma }_d(M)(x)}{d!}.$

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 $M$ there are cohomology classes $K_1,\dots ,K_s\in H^2(M,\mathbb{\mathbb{Z}})$, called Kronheimer–Mrowka basic classes (KM basic classes), as well as rational numbers $a_1,\dots ,a_s\in \mathbb{\mathbb{Q}}$, called Kronheimer–Mrowka coefficients (KM coefficients), so that:

${{𝒟}}_M(x)=\exp (Q_M(x,x)/2)\sum _{r=1}^s{a}_r\exp (K_r(x))$

for all $x\in H_2(M,\mathbb{\mathbb{Z}})$. Furthermore $w_2(M)=K_r\mathrm{mod}2\in H^2(M,{\mathbb{\mathbb{Z}}}_2)$ 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 $\mathrm{SW}:{\mathrm{Spin}}^{\mathrm{c}}(M)\to \mathbb{\mathbb{Z}}$ (hence the first Chern class $c_1:{\mathrm{Spin}}^{\mathrm{c}}(M)\to H^2(M,\mathbb{\mathbb{Z}})$ 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 $M$ with $b_2^{+}(M)\ge 2$ 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:

${{𝒟}}_M(x)=\exp (Q_M(x,x)/2)\sum _{\mathfrak{𝔰}\in {\mathrm{Spin}}^{\mathrm{c}}(M),\dim {{ℳ}}_{\mathfrak{𝔰}}^{\mathrm{S}\mathrm{W}}=0}{2}^{2+\frac{1}{4}(7\chi (M)+11\sigma (M))}\mathrm{SW}(M,\mathfrak{𝔰})\exp (c_1(\mathfrak{𝔰})(x)).$

References

• 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, doi:10.1090/S0273-0979-1994-00492-6, ISSN 0002-9904 
• 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, http://projecteuclid.org/getRecord?id=euclid.jdg/1214456482 
• Naber, Gregory L. (2011) (in en). Topology, Geometry and Gauge Fields. Applied Mathematical Sciences. 141 (Second ed.). Springer. doi:10.1007/978-1-4419-7895-0. ISBN 978-1-4419-7894-3. 

External links

• Kronheimer-Mrowka basic class at the nLab