## 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. Szabó [Topology Appl. 106, No. 3, 293–304 (2000; Zbl 0983.57026)] which is for the $$K3$$ alone.

### MSC:

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

### Citations:

Zbl 0983.57026; Zbl 1127.57303
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