His 1955 thesis at the University of Cambridge described a new theory termed "dihomology", an algebraic structure associated to a topological space, containing both homology and cohomology, introducing what is now known as the Zeeman spectral sequence.

This in turn led to the discovery of intersection homology by Robert MacPherson and Mark Goresky at Brown University where McCrory was appointed in 1974.

Zeeman is known among the wider scientific public for his contribution to, and spreading awareness of catastrophe theory, which was due initially to another topologist, René Thom, and for his Christmas lectures about mathematics on television in 1978.

He received an MA and PhD (the latter under the supervision of Shaun Wylie) from the University of Cambridge, and became a Fellow of Gonville and Caius College where he tutored David Fowler and John Horton Conway.

In his own words Zeeman's style of leadership was informal, but inspirational, and he rapidly took Warwick to international recognition for the quality of its mathematical research.

He was able to trade four academic appointments for funding that enabled PhD students to give undergraduate supervision in groups of two for the first two years, in a manner similar to the tutorial system at Oxford and Cambridge.

[7] Zeeman subsequently spent a sabbatical with Thom at the Institut des Hautes Études Scientifiques in Paris, where he became interested in catastrophe theory.

On his return to Warwick, he taught an undergraduate course in Catastrophe Theory that became immensely popular with students; his lectures generally were "standing room only".

[11] From these grew the Mathematics and Engineering Masterclasses for both primary and secondary school children that now flourish in forty centers in the United Kingdom.