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We construct a Hamiltonian Floer theory-based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of restriction maps, and prove some basic properties. Our main contribution is to identify natural geometric conditions in which relative symplectic cohomology of two subsets satisfies the Mayer–Vietoris property. These conditions involve certain integrability assumptions involving geometric objects called barriers — roughly, a –parameter family of rank coisotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (ie constant) Floer solutions are the main actors.
We establish a form of the h–principle for the existence of foliations of codimension at least which are quasicomplementary to a given one. Roughly, “quasicomplementary” means that they are complementary except on the boundaries of some kind of Reeb components. The construction involves an adaptation of W Thurston’s “inflation” process. The same methods also provide a proof of the classical Mather–Thurston theorem.
This is the first part of our project toward proving the Bershadsky–Cecotti–Ooguri–Vafa Feynman graph sum formula of all genera Gromov–Witten invariants of quintic Calabi–Yau threefolds. We introduce the notion of –mixed-spin- fields, construct their moduli spaces, their virtual cycles and their virtual localization formulas, and obtain a vanishing result associated with irregular graphs.
We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb, Wolfson and Wood (2019). Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil and Wood in 2015. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.
Given a three-dimensional Riemannian manifold containing a ball with an explicit lower bound on its Ricci curvature and positive lower bound on its volume, we use Ricci flow to perturb the Riemannian metric on the interior to a nearby Riemannian metric still with such lower bounds on its Ricci curvature and volume, but additionally with uniform bounds on its full curvature tensor and all its derivatives. The new manifold is near to the old one not just in the Gromov–Hausdorff sense, but also in the sense that the distance function is uniformly close to what it was before, and additionally we have Hölder/Lipschitz equivalence of the old and new manifolds.
One consequence is that we obtain a local bi-Hölder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson, Cheeger, Colding and Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally noncollapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.
Gromov and Lawson developed a codimension index obstruction to positive scalar curvature for a closed spin manifold , later refined by Hanke, Pape and Schick. Kubota has shown that this obstruction also can be obtained from the Rosenberg index of the ambient manifold , which takes values in the K–theory of the maximal –algebra of the fundamental group of , using relative index constructions.
In this note, we give a slightly simplified account of Kubota’s work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension submanifolds of Higson, Schick and Xie.
Given a triangulation of a closed surface, we consider a cross-ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross-ratio system induces a complex projective structure together with a circle pattern. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross-ratio systems with prescribed Delaunay angles to the Teichmüller space of the closed torus is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.
Let be the level- principal congruence subgroup of . Borel and Serre proved that the cohomology of vanishes above degree . We study the cohomology in this top degree . Let denote the Tits building of . Lee and Szczarba conjectured that is isomorphic to and proved that this holds for . We partially prove and partially disprove this conjecture by showing that a natural map is always surjective, but is only injective for . In particular, we completely calculate and improve known lower bounds for the ranks of for .
A consequence of the Cheeger–Gromoll splitting theorem states that for any open manifold of nonnegative Ricci curvature, if all the minimal geodesic loops at that represent elements of are contained in a bounded ball, then is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of escape from any bounded metric balls at a sublinear rate with respect to their lengths, then is virtually abelian.
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