We study the Seiberg-Witten invariant λSW(X) of smooth spin 4-manifolds X with integral homology of S1×S3 defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Frøyshov invariant h(X) and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology 3-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.

We make the elementary observation that the Lagrangian submanifolds of$C$^{$n$},$n$≥3, constructed by Ekholm, Eliashberg, Murphy and Smith are non-uniruled and, moreover, have infinite relative Gromov width. The construction of these submanifolds involve exact Lagrangian caps, which obviously are non-uniruled in themselves. This property is also used to show that if a Legendrian submanifold inside a contactisation admits an exact Lagrangian cap, then its Chekanov–Eliashberg algebra is acyclic.

We introduce a new geometric flow, called the chord shortening flow, which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord not orthogonal to the boundary would shrink to a point in finite time under the flow.

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We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of$g$copies of$S$^{$n$}×$S$^{$n$}, in the limit $${g \to \infty}$$ . Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.