A mathematical digression

and possibly a platitude. Recently, I discovered Bernard Cohn when I found a relatively cheap copy of his book "Colonialism and its forms of knowledge" in a book shop and found it very interesting. I then borrowed his "An anthrpologist among the historians" from the library and found it even more interesting. Some of the essays there seem to have become bases for later books by people like Nicholas Dirks and in one of the articles he suggests the choice of topics and methodolgy for research which seems cocrete and useful. But there is demand for all three copies of the book in the library and I was asked to return it before reading it completely. While travelling on the train to return it, I tried to see whether I could still think some mathematics which I have not been doing for a while. Surprisingly I could and it was about a couple of problems that I struggled with in the seventies and both solved now. The solutions suggest a conclusion which ia a platitude but it did not come to me naturally.
Both the problems are related to low dimensional topology and group theory. Before the days of Thurston and Perelman that was what most of us were doing. One of the questions is now a famous theorem "finitely generated three manifold groups are finitely presented" proved by Peter Scott in 1972 (independently by Peter Shalen but unpublished) and second is a theorem of Klaus Johannson that certain three manifolds are completely determined by their fundamental groups.
A group is like integers, both positive, negative and zero. Every element has an inverse (negative of it) so that if you add them, you get the identity element zero. Zero is called the identity since if you add it to any element, it does not change it. Any two elements can be added and addition is associative. Gadgets with these properties are called groups. But integers are a bit special in that the addition is commutative; whether you add a to b or b to a, you get the same result. This is because integers are just generated by one element 1, by adding it to itself repeatedly and taking negatives, one gets all the integers. If one takes gadgets like above with some addition operation which is associative, has identity and negatives, it is called a group. In general they are not commutative and if we start with some objects say a,b,c etc and form words freely and their negatives (usually called inverses when multiplicative notation is used) we get what is called a free group on the objects a,b,c etc. If there are n such, it is called free group on n generators and any group on n generators can be got from it by adding some relations. If the number of relations needed is finite (which we may not know before hand) the group is called finitely presented. Finitely presented groups can be represented by finite spaces and are usually more desirable than the complentary ones.
To show that finitely generated implies finitely presented for 3-manifold groups what one expects is a proposition of this type:
Prposition P: If G is finitely generated and indecomposable (as explained below), then the group obtained by adding sufficiently many finite number of relations is already indecomposable.
That finitely generated three manifold groups are finitely presented was proved in 1972 by Peter Scott (and independently by Peter Shalen, but unpublished). A key technical prposition involves decompositions of groups. The operation forming free groups from infinite cyclic groups generated by a,b,c etc above is called free product formation and the inverse operation 'a free product decomposition'. A group which cannot be decomposed into a free product is called 'indecomposable'. Any finitely generated group G can be written G=G1*G2*...*Gm*G(m+1)*...*G{m+n)where the last n factors are infinite cyclic and the first m factors are indecomposable but not infinite cyclic. The numbers m,n depend only on the group and the ordered pair (m+n,n) is called the complexity c(G) of the group G. A crucial technical proposition that Scott proves is:
Prposition Q: If there is a onto homomorphism from G to H which is one-to-one on the indecomposable factors of G other than the infinite cyclic ones, then c(G) is bigger than or equal to c(H) and equality holds if and only if the map is an isomorphiam (which means G,H are indistinguishable and are the same except in name. A homomorphism is a map which preserves the operations).
Somehow, this was not enough to prove Proposition P which is the natural approach to the original 3-manifold question and some more hard work and trickery was needed. Then after nearly 30 years, I met Thomas Delzant in a conference, and he told me during a lecture I could not follow that Proposition P was true. What all one had to do was to extend the notion of complexity to an onto homomorphism between groups, rather than just for a group.
A similar thing with respect to Johannson's theorem. In 1972, I found a simple proof by reducing it a group problem and showing that a certain group was trivial. There was an obvious relative version which was also useful but I could not prove. Again after 30 years, a colloborator (by some coincidence Peter Scott) insisted that I prove it and it turned out to be equivalent to saying that a cetain homomrphism of groups was trivial (every thing maps to zero).
Looking at both the problems, the crucial change seems to be that instead of looking at an object by itself, one had to look at a relation ( in these cases natural maps which are special cases of relations) between objects. Put this way, it seems to be a platitude but it did not occur to me for nearly 30 years.
It seems that I may have to keep doing mathematics until a copy of Bernard Cohn is available.