Embedded tori in four-manifolds
The authors give a product formula for certain \(SO(3)\) Donaldson invariants for four manifolds which split along a smoothly embedded 3-torus. Particular important cases are generalized logarithmic transformations \(X(\phi) = (X \smallsetminus nd(L)) \cup_\phi (D^2 \times T^2)\) and fiber sums \(Z(\phi) = (Z_1 \smallsetminus nd(L_1)) \cup_\phi(Z_2\smallsetminus nd(L_2))\) where the tori involved have self intersection zero. These generalize classical log transform and fiber sum for elliptic surfaces. As applications, Donaldson invariants for elliptic surfaces with \(b_2^+ = 1\) and odd multiplicities are computed, resolving a conjecture of Friedman, and Donaldson invariants are computed for \(F_g \times T^2\) for \(g \geq 1\) and \(SO(3)\) bundle \(P\) where \(w_2(P)\) is odd on \(T^2.\) There is also an analogous result for \(S^2 \times T^2\) which takes the chamber structure into account. Relative basic classes are defined for admissible smooth 4-manifolds \(X\) with \(\partial X = T^3\) and the Donaldson series of generalized log transforms and generalized fiber sums are given in terms of the relative basic classes. The authors, together with \textit{T. S. Mrowka}, have obtained analogous product formulas for Seiberg-Witten invariants [Math. Res. Lett. 4, No. 6, 915-929 (1997; Zbl 0892.57021)].
scientific article; zbMATH DE number 1307845