On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory (Q6595797)

This paper is concerned with determining the definite lattices arising from smooth \(4\)-manifolds without \(2\)-torsion in their homology that are bounded by certain homology \(3\)-spheres. Given an integer homology \(3\)-sphere \(Y\), definite lattices arising as intersection forms of smooth \(4\)-manifolds bounded by \(Y\) are determined.\N\NThe methods used are an extension of the instanton homology methods of Frøyshov. The results extend Donaldson's \textit{Theorem A}, which answers the above question for the \(3\)-sphere.\N\NThe main result is given in:\N\N\textbf{Theorem 1.1}. Let \(Y\) be an integer homology \(3\)-sphere \(\mathbb{Z}/2\)-homology cobordant to \(+1\) surgery on a knot with smooth \(4\)-ball genus \(2\). If a smooth, compact, oriented and definite \(4\)-manifold with no \(2\)-torsion in its homology has boundary \(Y\), then its intersection form is equivalent to one of\N\[\N\langle\pm1\rangle^n,\quad E_8\oplus\langle+1\rangle^n,\quad\Gamma_{12}\oplus\langle+1\rangle^n,\N\]\Nwhere \(\Gamma_{12}\) is the unique indecomposable unimodular positive-definite lattice of rank \(12\).\N\NThe proof can be found in Section 4. Section 2 contains the instanton inequalities that serve as the main tools in this paper, for example:\N\N\textbf{Theorem 2.5}. Let \(X\) be a smooth, compact, oriented \(4\)-manifold with homology \(3\)-sphere boundary \(Y\) and \(b_2^+(X)=n\geq1\). For \(1\leq i\leq n\), let \(\Sigma_i\subset X\) be smooth, orientable, connected surfaces in \(X\) of genus \(g_i\) with \(\Sigma_i\cdot\Sigma_i=1\), which are pairwise disjoint. Denote by \(\mathcal{L}\subset H^2(X;\mathbb{Z})/\text{Tor}\) the unimodular lattice of vectors vanishing on the classes \([\Sigma_i]\). Then\N\[\Nh(Y)+\sum_{i=1}^n \lceil\frac{1}{2}g_i\rceil\geq e_0(\mathcal{L}).\N\]\NTheir proofs can be found in Section 7. Section 3 contains the proof of the following:\N\N\textbf{Theorem 1.3}. Let \(Y\) be an integer homology \(3\)-sphere \(\mathbb{Z}/2\)-homology cobordant to \(+1\) surgery on a knot with smooth \(4\)-ball genus \(1\). If a smooth, compact, oriented and definite \(4\)-manifold with no \(2\)-torsion in its homology has boundary \(Y\), then its intersection form is equivalent to one of\N\[\N\langle\pm1\rangle^n,\quad E_8\oplus\langle+1\rangle^n.\N\]\NAn application is given in:\N\N\textbf{Corollary 3.4}. Let \(Y\) be an integer homology \(3\)-sphere \(\mathbb{Z}/2\)-homology cobordant to \(+1\) surgery on a knot with smooth \(4\)-ball genus \(1\). If a smooth, compact, oriented and definite \(4\)-manifold with no \(2\)-torsion in its homology has boundary \(Y\), then its intersection form is equivalent to one of\N\[\N\langle\pm1\rangle^n\text{ for some }n\geq1, \quad E_8\oplus\langle+1\rangle^n\text{ for some }n\geq0,\N\]\Nand all of these possibilities occur. \N\NSection 5 contains further examples of such applications.\N\NSection 6 contains a conjecture regarding the instanton cohomology of a circle times a surface. The cases for genus \(g\leq128\) are verified explicitly.\N\NSection 8 contains alternative proofs for some corollaries. These alternative proofs are more closely related to Frøyshov's methods. In particular, they make use of the instanton homology of \(3\)-spheres and Floer's exact triangle.\N\NSection 9 considers the example of a rank \(14\) definite unimodular lattice \(E_7^2\) in detail.