Graph complexes – a new tool in algebraic topology

Graph complexes – a useful tool in algebraic topology

Robin Stoll will receive his doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with professor Oscar Randal-Williams at the University of Cambridge.

Algebraic topology uses tools from algebra to investigate abstract geometric objects, such as a plane, a sphere or a torus (i.e. the surface of a donut). All three of these two-dimensional examples are so-called manifolds, which, together with analogues in higher dimensions, are the central objects of study in topology.

Over the past few centuries, the study of manifolds has played a vital role in various areas of mathematics and theoretical physics. However, many of their properties remain unknown, and the aim of this postdoctoral project is to explore new methods for understanding certain abstract manifolds with an odd number of dimensions. 

The question of classifying manifolds and their symmetries has driven the development of algebraic topology throughout the twentieth century. Two manifolds are said to belong to the same class if they can be deformed into each other without any tearing or glueing. Understanding the collection of all manifolds belonging to one such class is of particular importance. However, this problem has proved extremely difficult to tackle.
In this subject, it is important whether the dimension is even or odd. For manifolds in even dimensions, much progress was made over the last decade. The project will explore a new approach to investigate the odd dimensional case, using certain combinatorial structures called graph complexes.