Some remarks on research in Mathematics

Extract from a Preface to ‘Topology and Groupoids’, by Ronald Brown (2006) available from amazon.com I will be pleased if the exposition of this book can be improved in radical ways. Young mathematicians should be aware of the temporary mature of mathematical exposition. The attempt to form a ‘final view’ reminds one of the schoolboy question: what would happen if you laid worms in a straight line from Marble Arch to Picadilly Circus? Answer: one of them would be bound to wiggle and spoil it all. Some might argue that the groupoid worm has here not only wiggled from its accustomed place in topology, but become altogether too big for its boots, to which a worm, after all, has no rights. But I hope many will find it interesting to trace through this first attempt to answer, in part, the questions: Is it possible to rewrite homotopy theory, substituting the word groupoid for the word group, and making other consequential changes? If this is done, is the result more pleasing? These questions, both of the form ‘What if. . .?’, came to acquire for me a force and an obsession when pursued into the topic of higher homotopy groupoids. The scribbling of countless squares and cubes and their compositions lead to a conviction in 1966 that the standard group theory, once it was rephrased as a groupoid theory, had a generalisation to higher dimensions. Gradually, collaborations with Chris Spencer in 1971–1974, with Philip Higgins since 1974, with J.-L. Loday from 1981, and work of my research students, A. Razak Salleh, Keith Dakin, Nick Ashley, David Jones, Graham Ellis and Ghaffar Mosa, at Bangor, and Philip Higgins’ research students Jim Howie and John Taylor at London and Durham, made the theory take shape. In this way a worry of the algebraic topologists of the 1930s as to why the higher homotopy group were abelian, and so less complicated than the fundamental group, came to seem a genuine question. The surprising answer is that the higher homotopy groupoids are non-abelian, and are just right for doing many aspects of homotopy theory. In particular, they satisfy a version of the Van Kampen theorem which enables explicit and direct computations to be made. It will be interesting to see if the higher dimensional theory will come to bear a relation to the standard group theory similar to that of many-variable to one-variable calculus. But the higher dimensional theory is a story in which we cannot embark in this book. We now give the changes that have been made in this new edition. A section on function spaces in the category of k-spaces has been added the chapter 5. One of the reasons is that the material is quite difficult to find elsewhere. Another reason for its inclusion is that it will suggest to the reader that there are still matters to be decided on the appropriate setting for our intuitive notions of continuity. In any case, the generalisation from spaces to k-spaces makes the proofs if anything simpler. I am grateful to a number of people for comments, particularly Eldon Dyer and Peter May who suggested that chapter 7 needed clarification, and Daniel Grayson who suggested the notation [(X, i), (Y, u)] now used in chapter 7 to replace the original X//u, which was non-standard and too brief. (But this double slash is used in a new context in chapter 9.) In the event, chapter 7 has been completely revised to make use of the term cofibration rather than HEP. The idea of fibration of groupoid, which came to light towards the end of the writing of the first edition and so appeared there only in an exercise, has now been incorporated into the main text. However, I have not included fibrations of spaces, since to do so would have enlarged the text unduly, or force the omission of material for which no other textbook account is available. In chapter 8, an error in section 8.2 has been corrected. Also free groupoids have now been used to define the notion of path in a graph. An exercise on the computation of the fundamental group of a union of non-connected spaces has now been incorporated into the text, as an illustration of the methods and because of its intrinsic importance. This result is used in a new section which gives a proof of the Jordan Curve Theorem, and some new results on the Phragmen-Brouwer Property. In chapter 9, section 9.4 on the existence of covering groupoids has been rewritten to give a clearer idea of the notion of action of groupoids on sets. Section 9.5 includes some results on topologising the fundamental groupoid. The relation between covering spaces of X and covering groupoids of πX has been clarified by adding section 9.6, which gives an account of the equivalence of the categories of these objects. This enables an algebraic account of the theory of regular covering spaces. Section 9.7 gives a new account of pullbacks of covering spaces and covering morphisms, using exact sequences. Section 9.8 gives an account