2019-07-04

Paper of the Week

After the early 1980s, gauge theory has been used by mathematicians to study 4-dimensional spaces (manifolds). The hottest mathematical subject studied by gauge theory is a sort of difference between the shape of topological (continuous) 4-dimensional manifold and that of smooth 4-dimensional manifold. On the other hand, in most of mathematical areas, it is quite significant to understand symmetry of a given mathematical object, such as a manifold. However, about the space of all symmetries of a 4-dimensional manifold, only few things have been known. In this paper, we study the spaces of symmetries of 4-dimensional manifolds. We revealed that gauge theory (more precisely Seiberg-Witten theory) can extract some difference between the shape of the space of symmetries of a topological 4-dimensional manifold and that of a smooth 4-dimensional manifold. In addition to gauge theory, we also used some classical but deep results in higher-dimensional topology invented 50 years ago. Such a combination of gauge theory with classical results in topology is also a new and interesting direction of this paper.

Reference
Tsuyoshi Kato, Hokuto Konno, Nobuhiro Nakamura
"Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds"
arXiv: 1906.02943