Mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation (Q2474976)
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| English | Mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation |
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Mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation (English)
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20 March 2008
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\textit{D. Kotschick} conjectured in [Math. Ann. 292, No.~2, 375--381 (1992; Zbl 0753.14034)] that for a closed simply-connected 4-manifold all Donaldson (or Seiberg-Witten) invariants vanish for one of its two orientations. While, as the author concedes, he did not find new examples for which the conjecture was unknown, he has been able to prove a mod 2 vanishing result for the Seiberg-Witten invariants of certain 4-manifolds which admit an involution. This can be applied to some complex surfaces with an anti-homolomorphic involution, but, as the author points out, both examples discussed can also be treated using other results. As for the method of proof, the author uses a version of the Bauer-Furuta stable homotopy refinement of the Seiberg-Witten invariants, as these are known to allow for equivariant (and parametrised) generalisations. It remains unclear why the references mostly list unpublished preprints in places where published accounts exist. For a start, see [\textit{S. Bauer} and \textit{M. Furuta}, Invent. Math. 155, No.~1, 1--19 (2004; Zbl 1050.57024)], [\textit{S. Bauer}, Different faces of geometry, S. Donaldson et al. (ed.), New York, NY: Kluwer Academic/Plenum Publishers, 1--46 (2004; Zbl 1083.57039)], and the references therein.
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