Donaldson’s polynomial invariant \(q_{l,M}\) is defined for a closed oriented simply connected 4-manifold with \(b_ 2^ +\) odd \(>1\) and sufficiently large \(l.\) When \(b_ 2^ +\) is even, one has \(\dim {\mathcal M}_{M,k}=2d+1\); if \(k\) is even and \(M\) is spin, there is a nontrivial class \(u_ 1 \in H^ 1({\mathcal B}^*_{M,k};\;\mathbb{Z}_ 2)\). This allows one to define a polynomial invariant \(q_{k,u_ 1,M}\) of degree \(d\) in \(H_ 2(M;\mathbb{Z}_ 2)\) with values in \(\mathbb{Z}_ 2\) which is an invariant of the smooth structure of \(M\). Similarly, when \(b^ +_ 2\) is odd \(>1\), \(\dim {\mathcal M}_{N,k}=2d+2,k\) odd and \(N\) spin, there is a nonzero class \(u_ 2 \in H^ 2({\mathcal B}_{N,k}; \mathbb{Z}_ 2)\), which allows definition of a degree \(d\) polynomial \(q_{k,u_ 2, N}\) invariant with values in \(\mathbb{Z}_ 2\). In this paper the authors study these two 2- torsion invariants and relate them to each other and to the usual Donaldson invariant. In particular, stabilization formulas are proved where hypotheses allowing definitions of these invariants hold: \[ \begin{aligned} q_{l,M} (z_ 1,\dots,z_ d) & \equiv q_{l+1,u_ 1,M \# S^ 2 \times S^ 2} (z_ 1,\dots,z_ d,x,y) \text{mod} 2 \\ q_{k,u_ 1,N} (z_ 1,\dots,z_ d) & \equiv q_{k+1,u_ 2,N \# S^ 2 \times S^ 2} (z_ 1,\dots,z_ d,x,y) \text{mod} 2 \end{aligned} \] where \(x=[S^ 2 \times 0]\), \(y=[0 \times S^ 2]\).
As consequence of these formulas, it is shown that whenever \(M_ 1\) and \(M_ 2\) are homotopy equivalent closed simply connected smooth spin 4- manifolds (so homeomorphic) and \(M_ 1 \# k(S^ 2 \times S^ 2)\) is diffeomorphic to \(M_ 2 \# k(S^ 2 \times S^ 2)\), \(k=1,2\), then \(q_{l,M_ 1}\) and \(q_{l,M_ 2}\) have the same parity. Wall has shown that for any 2 homotopy equivalent smooth simply connected closed 4-manifolds, there must be some \(k\) so have \(M_ 1 \# k(S^ 2 \times S^ 2)\) is diffeomorphic to \(M_ 2 \# k(S^ 2 \times S^ 2)\). It is an unsolved question as to what the minimal such \(k\) is. There are many manifolds (e.g. simply connected elliptic surfaces) where \(k=1\) is sufficient, and there are currently no known examples requiring \(k \geq 2\). If one could show that two Donaldson polynomials \(q_{l,M_ 1}\), \(q_{l,M_ 2}\) had different parity, then this would provide an example where \(k \geq 3\). However, the authors show that the Donaldson polynomial usually vanishes mod 2 for odd \(l\) for closed simply connected spin 4- manifolds, and so such examples may be difficult to find.
The primary technique used is a Mayer-Vietoris argument developed in the thesis of T. Mrowka and in work of C. Taubes and J. Morgan, T. Mrowka, and D. Ruberman. This is a highly important but technically difficult method, and the paper is recommended for the exposition of its application here.