Dr. John Nicholson

Algebraic topology @ Imperial College London

Dr. John Nicholson

The topology of 2-complexes

It has long since been known that techniques developed to study the topology of high dimensional manifolds do not suffice for manifolds in dimension 4. One reason for this is that the topology of 4-manifolds is closely related to the topology of 2-dimensional cell complexes, and hence to hard open problems concerning group presentations such as the Andrews-Curtis Conjecture and the Relation Gap Problem.

In the 1970s, C. T. C. Wall compiled a list of eight problems concerning the topology of 2-complexes. It is widely believed that there exist counterexamples to most, if not all, of them, though it took until 1990 for one such example to be found.

The aim of this talk will be to give a light introduction this topic and to discuss some of my recent work which includes a counterexample to another one of the problems on Wall's list. A surprising feature is how much the techniques needed can vary depending on the choice of (fundamental) group G. This ranges from algebraic number theory when G is finite, to geometric group theory when G is infinite.