Search

We prove that it is relatively consistent with \mathsf {ZFC} that every maximal ideal is Katětov above of all F_\sigma ideals. In particular, we prove that it is consistent that there is no Hausdorff ultrafilter. The main theorem answers questions from Mauro Di Nasso and Marco Forti [Proc. Amer. Math. Soc. 134 (2006), pp. 1809–1818], Jana Flašková [WDS'05 proceedings of contributed papers: part I - mathematics and computer sciences, 2005; Comment. Math. Univ. Carolin. 47 (2006), pp. 617–621; 10th Asian logic conference, World Sci. Publ., Hackensack, NJ, 2010], Osvaldo Guzmán and Michael Hrušák [Topology Appl. 259 (2019), pp. 242–250], and Mauro Di Nasso and Marco Forti [Logic and its applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 2005], and gives a different model for a question from Michael Benedikt [J. Symb. Log. 63 (1998), pp. 638–662], which was originally solved by S. Shelah [Logic colloquium '95 (Haifa), lecture notes logic, Springer, Berlin, 1998]. [ABSTRACT FROM AUTHOR]

Introduced by Seifert and Threlfall [ Variationsrechnung im Grossen , Hamburger Math Einzelschr., Heft 24, Teubner, Leipzig-Berlin, 1938], cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik-Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate. If, contrary to the assertion of the conjecture, there are isolated critical points that do not admit cylindrical ball neighborhoods, then we say that such critical points are exotic. We prove a Lusternik-Schnirelmann type theorem asserting that the minimal number of critical points of smooth functions without exotic critical points on a closed manifold of dimension at least 6 is the same as the minimal number of elements in a Singhof-Takens filling of M by smooth balls with corners. [ABSTRACT FROM AUTHOR]

We study the symplectic mapping class groups of (CP2\#5CP2,ω). Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures J_0 as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by Wu [Math. Ann. 359 (2014), pp. 153–168]) and the study of a semi-toric model to understand a "connecting map", whose cokernel is the symplectic mapping class group. Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms \omega. Let N_{\omega } be the number of (−2)-symplectic spherical homology classes. Torelli SMCG is trivial if N_{\omega } > 8; it is \pi _0(Diff^+(S^2,5)) if N_{\omega } = 0 (by Seidel [ Lecture notes in Math. , Springer, Berlin, 2008] and Evans [J. Symplectic Geom. 9 (2011), pp. 45–82]); and it is \pi _0(Diff^+(S^2,4)) in the remaining case. Further, we completely determine the rank of \pi _1(Symp(C P2\#5C P2,\omega)) for any given symplectic form. Our results can be uniformly presented in terms of Dynkin diagrams of type A and type D Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds (open problem 16 in McDuff-Salamon's book 3rd version [ Oxford graduate texts in mathematics , Oxford University Press, Oxford, 2017]). [ABSTRACT FROM AUTHOR]