A gluing theorem for the relative Bauer-Furuta invariants. (English) Zbl 1128.57031

Previously in [Periodic Floer pro-spectra from the Seiberg-Witten equations, math.GT/0203243 and Geom. Topol. 7, 889–932 (2003; Zbl 1127.57303)], the author and P. Kronheimer have defined the relative Bauer-Furuta invariant \(\Psi\) for smooth 4-manifolds with boundary. By using this relative invariant, the current paper establishes a gluing theorem for the original Bauer-Furuta invariant of a closed 4-manifold \(X\) that is decomposable into two manifolds \(X_1, X_2\) along a common boundary \(Y\) (which is assumed to be a rational homology 3-sphere).
The main idea is to apply the finite dimensional approximation to \(X, X_1, X_2\) simultaneously. When \(X\) is a cobordism between \(Y_1\) and \(Y_2\), the invariant \(\Psi(X)\) restricts to a morphism \({\mathcal D}_X\) between the Seiberg-Witten Floer spectra SWF\((Y_1)\) and SWF\((Y_2)\) with a possible change of degree. The second main result in the paper gives a gluing theorem for \({\mathcal D}_X\) in the new context. Finally as applications, the auther shows the connected sum \(K3\#K3\#K3\) does not split off any exotic nucleus \(N(2)_{p,q}\), for \(p,q\geq1\) relatively prime. This extends a previous theorem of A. Stipsicz and Z. Szab√≥ [Topology Appl. 106, No. 3, 293–304 (2000; Zbl 0983.57026)] which is for the \(K3\) alone.


57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology
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