by Peter Kropholler
Peter’s survey article in the Bulletin of the LMS1 is widely known. For me it was a vital tool to help me understand the basics of Thurston’s Geometrization Conjecture and at the time I was looking at this in the late nineteen eighties, it was one of the few accessible sources for a reader not greatly familiar with the geometrical background or even the theory of Lie groups. It formed part of a platform on which I could stand in order to contemplate algebraic versions of the JSJ torus decomposition theorem. Nowadays these have been taken forward far beyond what I envisaged by many authors, notably including Guirardel and Levitt and also of course Scott and Swarup.
Peter was a gifted and patient teacher: he once explained to me in layman’s terms why one would expect a four-dimensional PL manifold to have a smooth structure but not a seven- or eight-dimensional PL manifold. From his explanation it was at once transparent why the theory of smooth and PL manifolds should be expected to diverge in high dimensions and it was clear why cohomology of classifying spaces based on certain exotic cohomology theories would be relevant in calculating what was going on. I am greatly indebted for that: it was one of the most significant learning experiences I had in my postdoctoral years.
For a while Peter dismissed my attempts at an algebraic torus theorem on grounds that a key property of centralizers in 3-manifolds that I relied on required the geometry I was claiming to replace by algebra. But a few years later I had a version that Peter agreed quite vocally met the criteria to be called an algebraic version. I am grateful very much for that because I think it helped to have my work recognised. As mentioned above, others took it much further as my research moved in different directions.
So although I only met Peter a small handful of times, it turned out that he was very influential.
