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4-manifolds and intersection forms with local coefficients. (English) Zbl 1257.57033

The first application of gauge theory to low-dimentional topology was Donaldson’s theorem that any smooth negative-definite \(4\)-manifold has a standard intersection form. The original proof related the number of singularities of the ASD moduli space to the ring structure of the cohomology of the manifold. In this paper Frøyshov establishes the following generalization: if a homology \(3\)-sphere bounds a \(4\)-manifold with non-standard intersection form on cohomology with twisted coefficients and meeting some extra homological conditions then it has a non-trival homomorphism from the fourth instanton Floer homology group to \(\mathbb{Z}_2\) defined by counting instantons connecting a flat connection to the trivial connection. One novel ingredient in this work is counting reducibles with stabilizer \(\mathbb{Z}_2\) as opposed to the usual \(\text{U}(1)\). Another innovation is to consider double coverings of a subset of the moduli of connections.

MSC:

57R57 Applications of global analysis to structures on manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R58 Floer homology
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