- ДокументFloer-Novikov cohomology and symplectic fixed points, revisited(2020) Kaoru Ono, Hong Van LeThis note is mostly an exposition of a few versions of Floer-Novikov cohomology with a few new observations. For example, we state a lower bound for the number of symplectic fixed points of a non-degenerate symplectomorphism, which is symplectomorphic isotopic to the identity, on a compact symplectic manifold, more precisely than previous statements in [14,10].
- ДокументOlympic links in a Chebotarev link(2020) Jun UekiThe Chebotarev law for an infinite link is an equidistribution property about how its components are linked in a group theoretic sense. We overview several properties of a Chebotarev link following the author's article "Chebotarev links are stable generic". In addition, we exhibit the density of modulo 2 Olympic links in a Chebotarev link.
- ДокументOn the generalization of Inoue manifolds(2020) Andrei Pajitnov, Endo HisaakiThis paper is about a generalization of celebrated Inoue's surfaces. To each matrix M in SL(2n+1,ℤ) we associate a complex non-Kähler manifold TM of complex dimension n+1. This manifold fibers over S1 with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.
- ДокументOn Rham cohomology of locally trivial Lie groupoids over triangulated manifolds(2020) Jose R. OliveiraBased on the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology of a transitive Lie algebroid, it is proved that the Rham cohomology of a locally trivial Lie groupoid G on a smooth manifold M is isomorphic to the piecewise Rham cohomology of G, in which G and M are manifolds without boundary and M is smoothly triangulated by a finite simplicial complex K such that, for each simplex ∆ of K, the inverse images of ∆ by the source and target mappings of G are transverses submanifolds in the ambient space G. As a consequence, it is shown that the piecewise de Rham cohomology of G does not depend on the triangulation of the base.
- ДокументA survey of homotopy nilpotency and co-nilpotency(2020) Marek GolasinskiWe review known and state some new results on homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of homotopy nilpotency of homogenous spaces G/K for a Lie group G and its closed subgroup K < G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is shown.