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Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz num- ber of the map induced by this involution in the reduced mono- pole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic ar- gument, making use of an exact triangle in monopole Floer homol- ogy, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants as- sociated to spin structures on Y . We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology. 

In this paper, we use the [Formula: see text]-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic [Formula: see text]-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the [Formula: see text]-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold. 

Abstract A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $$\mathbb R\rm{P}^1$$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres.