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Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations. (English) Zbl 1001.58004

The paper gives a computation of the virtual dimensions of finite-energy Seiberg-Witten moduli spaces for 4-manifolds bounding unions of Seifert fibrations. This requires to determine the Atiyah-Patodi-Singer index near the boundary. The author points out that the relevant operator here is not just a Dirac operator but its sum with an odd signature operator. A main body of the paper is devoted to the computations of the eta function of this operator. As applications of these computations, the author further obtains upper estimates for the Froyshov invariants of many Brieskorn homology spheres, which in turn lead to topological results that any negative definite 4-manifolds bounding some of these 3-manifolds must have a diagonalizable intersection form.
In the special case of cylinders over Seifert fibrations, the virtual dimension was previously computed by T. Mrowka, P. Ozsvath and B. Yu [Commun. Anal. Geom. 5, No. 4, 685-791 (1997; Zbl 0933.57030)] using a more algebraic-geometry approach. The author notes that his formula looks quite different from that of Mrowka-Ozsvath-Yu; it would be interesting to verify directly that the two descriptions are equivalent.

MSC:

58D27 Moduli problems for differential geometric structures
58J20 Index theory and related fixed-point theorems on manifolds
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58J28 Eta-invariants, Chern-Simons invariants

Citations:

Zbl 0933.57030
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