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Equivariant Bauer-Furuta invariants on some connected sums of 4-manifolds. (English) Zbl 1421.57041
Summary: On some connected sums of 4-manifolds with natural actions of finite groups, we use equivariant Bauer-Furuta invariant to deduce the existence of solutions of Seiberg-Witten equations invariant under the group actions.
For example, for any integer \(k\geq 2\) we show that the connected sum of \(k\) copies of a 4-manifold \(M\) with nontrivial Bauer-Furuta invariant has a nontrivial \(\mathbb{Z}_k\)-equivariant Bauer-Furuta invariant for the obviously glued Spin\(^c\) structure, where the \(\mathbb{Z}_k\)-action cyclically permutes \(k\) summands of \(M\). This contrasts with the fact that ordinary Bauer-Furuta invariants of such connected sums are all trivial for any sufficiently large \(k\), when \(b_1(M)=0\).
MSC:
57R57 Applications of global analysis to structures on manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions
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