The New Topology

The next step in Brouwer’s topological research was the study of continuous maps on manifolds. The program opened with a bang: in a brief note Brouwer proved the invariance of dimension under homeorphisms. This publication led to an unpleasant altercation with Lebesgue, who claimed to have already found a proof. In fact he had deduced the invariance from the paving principle, but failed to prove the paving principle. In the end Brouwer’s priority and superior insight was fully vindicated. In subsequent papers Brouwer enriched the arsenal of basic notions of topology with simplicial approximation and the mapping degree. The contacts with Baire, Hadamard, Blumenthal, and Hilbert, are described. Brouwer’s name became lastingly attached to his fixed point theorem. Brouwer also proved the invariance of domain theorem, which he subsequently used to salvage Klein’s continuity method for proving uniformisation. This brought him into a conflict with Paul Koebe, who was the uncrowned king of uniformisation and complex function theory. This first topological period closed with a significant feat: Brouwer defined, following Poincaré’s first approach, the general notion of dimension, and proved its ‘correctness’, i.e. showed that ℝ n is n-dimensional.