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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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