Ikegami and Saeki have proved that the cobordism group of Morse functions on oriented surfaces is an infinite cyclic group. Their method is applicable with a little modification to the computation of the cobordism group of Morse functions on unoriented surfaces. We prove that this group is isomorphic to the direct sum of the infinite cyclic group and the finite group of order two.

This paper uses ALIS data to compare academic performance in two subjects often viewed as relatively close substitutes for one another at A-level. The important role of GCSE achievement is confirmed for both subjects. There is evidence of strong gender effects and variation in outcomes across Examination Boards. A counterfactual exercise suggests that if the sample of Business Studies candidate...

We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further that imply they are ambiently isotopic. Along way, we prove certain pairs of group, homeomorphic boundaries, equivalent equivariant int...

Abstract We give a characterisation of Fano-type surfaces with large cyclic automorphisms. As an application, we Kawamata log terminal $3$ -fold singularities class groups rank at least $2$ .

Consider the moduli space M g of Riemann surfaces of genus g ≥ 2 and its Deligne-Mumford compactification M g. We are interested in the branch locus B g for g > 2, i.e., the subset of M g consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in M g but the set of (cyclic) trigonal surfaces is not. By contrast, t...

In the first part, we give a self contained introduction to the theory of cyclic systems in n-dimensional space which can be considered as immersions into certain Grassmannians. We show how the (metric) geometries on spaces of constant curvature arise as subgeometries of Möbius geometry which provides a slightly new viewpoint. In the second part we characterize Guichard nets which are given by ...

A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconne...