Connections, cohomology and the intersection forms of 4-manifolds.

*(English)*Zbl 0635.57007Beside 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.

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.) |