224 A History ofMathematics
Fig. 3Circle, torus, and sphere.Fig. 4A torus and a knotted torus (which, incidentally are linked), are homeomorphic.Again, this was very unlike Göttingen mathematics. It looked forwards (nothing like it had been
done before) and backwards (it went against the growing current of abstraction and the need for
certainty). How was one to be sure that two different cell decompositions gave the same numbers?
It took some time for a satisfactory proof to emerge, but in the meanwhile the handful who were
interested in topology were happy to make a start with Poincaré’s ideas and methods, and his
‘invariants’. And, interestingly and importantly, his main followers came not from France, but from
America (Veblen, and then Alexander) and Germany (Dehn, Heegaard, and then Reidemeister).
Under their influence, and that of the Russian Alexandrov, a close associate of Noether, topology
in the 1920s and 1930s became ‘algebraic topology’. In discussions with Noether, Alexandrov
realized that Poincaré’s invariants concealed a group.^8 What had been the intuitive subject par
excellence had been forced to define itself. As we shall see, worse was to come. All the same, as
we meet it in Seifert and Threlfall’s classicLehrbuch der Topologieof 1934 (reprinted as 1980), the
language may seem strange, and almost all of today’s methods are missing (homotopy groups,
exact sequences, fibre spaces,...) but the basic objects are in place.^9 What perhaps most strikes
today’s reader is that theLehrbuchis packed full of attractive and illuminating pictures (see Fig. 5).
As an image of the kind of work that topologists do (cutting out knots and gluing them in with
a twist, say), they are an invitation to read further, even though the text is not always easy. While
topology was a great deal more abstract, it was still the most graphic of mathematical pursuits.- There is an account of this in Corry (2004, p. 245), with mention of a possible claim by Mayer.
- As time went on higher-dimensional applications became more important, so that Seifert-Threlfall is still the ‘easiest’ reference
for two and three dimensions.