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The authors prove an analogue of the Kotschick-Morgan Conjecture in the context of \\mathrm{SO(3)} monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the \\mathrm{SO(3)}-monopole cobordism. The main technical difficulty in the \\mathrm{SO(3)}-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible \\mathrm{SO(3)} monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of \\mathrm{SO(3)} monopoles. In this monograph, the authors prove--modulo a gluing theorem which is an extension of their earlier work--that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. Their proofs that the \\mathrm{SO(3)}-monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze and Witten's Conjecture in full generality for all closed, oriented, smooth four-manifolds with b_1=0 and odd b^+\\ge 3 appear in earlier works.

It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.

We compute the $\\text{Pin}(2)$ -equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\\unicode[STIX]{x1D6FD}=-\\bar{\\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\\unicode[STIX]{x1D6FC},\\unicode[STIX]{x1D6FD},$ and $\\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\\unicode[STIX]{x1D6F4}(a_{1},\\ldots ,a_{n})$ are not homology cobordant to any $-\\unicode[STIX]{x1D6F4}(b_{1},\\ldots ,b_{n})$ . We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\\text{Pin}(2)$ -equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\\unicode[STIX]{x1D6FC},\\unicode[STIX]{x1D6FD},$ and $\\unicode[STIX]{x1D6FE}$ . In particular, we identify an $\\mathbb{F}[U]$ -module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

This paper introduces a new Lagrangian surgery construction that generalizes Lalonde–Sikorav and Polterovich’s well-known construction, and combines this with Biran and Cornea’s Lagrangian cobordism formalism. With these techniques, we build a framework which both recovers several known long exact sequences (Seidel’s exact sequence, including the fixed point version and Wehrheim and Woodward’s family version) in symplectic geometry in a uniform way, and yields a partial answer to a long-term open conjecture due to Huybrechts and Thomas; this also involved a new observation which relates projective twists with surgeries.

We define a graph encoding the structure of contact surgery on contact $3$ -manifolds and analyse its basic properties and some of its interesting subgraphs.

We study the equivariant cobordism rings for the action of a torus T on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational T -equivariant cobordism rings of smooth projective G -spherical varieties with the action of a maximal torus T of G . As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.