부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title: Homology Manifolds and $g_2$

Abstract: The $g$-vector of a simplicial complex contains significant information about the combinatorial and topological structure of the complex. Several classification results concerning the structure of normal pseudomanifolds and homology manifolds have been established in relation to the value of $g_2$, the third component of the $g$-vector. It is known that when $g_2=0$, all normal pseudomanifolds of dimensions at least three are stacked spheres. Walkup proved that a homology $3$-manifold with $g_2\leq 9$ is a triangulated sphere. In this talk, we demonstrate that for $d\geq 3$, the homology $d$-manifolds with $g_2\leq 3$ are triangulated spheres. In particular, if $d=4$, then this bound can be extended until $g_2\leq 5$, and it is the best possible bound in dimension $4$ for homology manifolds being triangulated spheres.

Title: Hirzebruch genera and rigidity equations

Abstract: A Hirzebruch genus is a ring homomorphism from the complex or oriented bordism ring to some ring R. Any such homomorphism can be extended in the standard way to a homomorphism from the corresponding bordism ring of manifolds with a k-dimensional torus action to the power series ring R[[x_1, ..., x_k]]. A genus is called rigid on a T^k-manifold M if the value of its extension on the bordism class of M is a constant series. The rigidity of genera has been intensively studied in the works of V. Buchstaber, T. Panov and N. Ray, and in many cases this property is equivalent to the multiplicativity of a genus with respect to fiber bundles with fiber M. There is a localization formula expressing the value of the equivariant extension of a genus in terms of the fixed points data of M. Using this formula, the property of rigidity is written as a functional equation for the exponential of a genus. I will give a brief introduction and discuss some of the results obtained using this approach.