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Connections, cohomology and the intersection forms of 4-manifolds. (English) Zbl 0635.57007
Beside Freedman’s results the following well known fundamental theorem of the author initiated the revolution in 4-manifold theory. Theorem A. If $$X^ 4$$ is smooth, compact, and simply connected and with negative intersection form $$(\alpha\cdot\alpha \leq 0$$ for all $$\alpha \in H_ 2)$$, then the form is equivalent over the integers to the standard form $$(-1)\oplus \cdot \cdot \cdot \oplus (-1).$$ In this paper the author discusses the existence of smooth, simply connected 4-manifolds with certain indefinite intersection forms. Theorem B. If $$X^ 4$$ is a smooth, simply connected, spin 4-manifold whose intersection form has one positive part, then the form is equivalent over the integers to $$\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}.$$
Theorem C. If the intersection form of such a manifold has two positive parts, then it is equivalent over the integers to $$\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix} \otimes \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$$. Corollary. The classical Kummer K3 surface K is smoothly indecomposable. In the earlier proof of theorem A the author constructed the moduli spaces of anti self- dual gauge fields using concentrated or particle-like connections. This moduli space provided a cobordism between a definition manifold $$X^ 4$$ and a number of copies of $$P^ 2(C)$$, and the theorem was deduced from the cobordism invariance of signature. The proof of theorems B, C is shifted from cobordism to homology. If (.,.) is an even, unimodular form on a lattice L of rank r, then one can associate to its mod 2 reduction on $$L\otimes {\mathbb{Z}}/2$$ a form $$\omega \in \Lambda^ 2(L^*\otimes {\mathbb{Z}}/2)$$, $$\omega (\alpha_ 1,\alpha_ 2)=\alpha_ 1\cdot \alpha_ 2$$ mod 2. r has to be even, $$r=2p$$ and $$\omega^ p\in \Lambda^ r(L^*\otimes {\mathbb{Z}}/2)$$ is nonzero. $$\omega^ d=0$$ impies $$d>p$$. Defining $$Q_ 4(\alpha_ 1,\alpha_ 2,\alpha_ 3,\alpha_ 4)=(\alpha_ 1\cdot \alpha_ 2)(\alpha_ 3\cdot \alpha_ 4)+(\alpha_ 2\cdot \alpha_ 3)(\alpha_ 2\cdot \alpha_ 4)+(\alpha_ 1\cdot \alpha_ 4)(\alpha_ 2\cdot \alpha_ 3)$$ mod 2 and $$Q_ 6(\alpha_ 1,...,\alpha_ 6)=\sum (\alpha_{i_ 1}\cdot \alpha_{i_ 2})(\alpha_{i_ 3}\cdot \alpha_{i_ 4})(\alpha_{i_ 5}\cdot \alpha_{i_ 6})$$ mod 2 one gets $$r\leq 2\Leftrightarrow Q_ 4=0$$ and $$r\leq 4\Leftrightarrow Q_ 6=0.$$
Now the author considers some space of multi-instanton solutions introduced by Taubes and associates to each homology class in $$X^ 4$$ a cohomology class in this space. Considering the cup product of these cohomology classes, the expressions $$Q_{2d}$$ appear as the number of boundary components of multi-instanton solutions. Assuming $$b^+_ 2(X^ 4)=1$$ resp. $$=2$$, the author deduces theorems B, C. As a main difference to the earlier considered concentrated connections now there appear extra parameters, detected by torsion classes using the index of a Dirac family.
Reviewer: J.Eichhorn

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R20 Characteristic classes and numbers in differential topology 53C05 Connections, general theory 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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