Open manifolds, Ozsváth-Szabó invariants and exotic \(\mathbb R^4\)'s (Q984624)

The author defines a differential topology invariant for open 4-manifolds, end Floer homology \(HE(X,{\mathfrak s})\), and shows that the invariant is non-trivial. The homology \(HE(X,{\mathfrak s})\) is an abelian group valued invariant and \(HE({\mathbb R}^4,{\mathfrak s})=\{0\}\). Suppose that an open 4-manifold is given by an exhaustion of compact manifods \[ K_1\subset K_2\subset \cdots\subset K_i\subset \cdots=X. \] The cobordidms between \(K_i\) and \(K_j\) \((i<j)\) induce the homomorphisms \(F_{K_j-K_i}:HF_{\text{red}}(K_i,{\mathfrak s}_i)\to HF_{red}(K_j,{\mathfrak s}_j)\). The reduced Heegaard Floer theory was defined in [\textit{P. Ozsváth} and \textit{Z. Szabó}, Ann. Math. (2) 159, No. 3, 1027--1158 (2004; Zbl 1073.57009)]. Taking the direct limit of the directed system of these abelian groups, the author defines end Floer homology. However for the well-definedness the cobordisms require admissibility conditions for any cobordism between \(K_i\) and \(K_j\) for \(i<j\). For example a 1-handle is admissible and a 2-handle whose attaching circle is a primitive, non-torsion homology class in the 3-manifold is admissible. To prove the non-triviality of end Floer homology, the author prepares an open 4-manifold \(X=Y\cup N\times[0,1]\cup Q\sim {\mathbb R}^4\) (homeomorphism), where \(N\) is a contact 3-manifold and \(Y=K_1\) is obtained by surgery along a sphere in \(Y'\) that is a 4-ball attached by a 2-handle along a non-trivial slice knot. \(Q\) is a symplectic manifold, whose end is a Casson tower. The manifold \(X\) can be seen as an admissible exhaustion. There exists a non-trivial element \(z_1\in HF_{\text{red}}(\partial Y ,t_1)\). The maps \(F_{K_j-K_1}\) are non-zero, since the cobordisms in between are symplectic, so the maps make the images of \(z_1\) non-zero. This implies \(EH(X)\neq 0\).