The end of the paper “Finite H-spaces and Lie groups” by J.F. Adams (Journal of Pure and Applied Algebra, vol. 19 (1980)) contains the following, amazing purported “Letter from E8”.
Mathematicians may be divided into two classes; those who know and love Lie groups, and those who do not. Among the latter, one may observe and regret the prevalence of the following opinions concerning the compact exceptional simple Lie group of rank 8 and dimension 248, commonly claled E8.
- That he is remote and unapproachable, so that those who desire to make his acquaintance are well advised to undertake an arduous course of preparation with E6 and E7.
- That he is secretive; so that any useful fact about him is to be found, if at all, only at the end of a long, dark tunnel.
- That he holds world records for torsion.
Point (1) deserves the following comment. Any right-thinking mathematician who wishes to construct the root-system of E6 does so as follows: first he constructs the root-system of E8, and then inside it he locates the root-system of E6. In this way he benefits from the great symmetry of the root-system of E8, and its perspicuous nature. If this good precedent is not followed in their researches, one should consider whether to infer a lack of boldness in the investigator rather than a lack of cooperation from the subject-matter.
Since point (2) is equivalent ot point (1), we may pass to point (3). And here we should first reject the defences offered by some who might otherwise pass as well-informed. For they appear to regard it as a venial blemish on an otherwise worthy character, comparable to holding world records for the drinking of beer. This will not do. Let us first consider the rioutous profusion of torsion displayed by such groups as PSU(n). It then becomes clear that one can award a title to E8 only be restricting the competition to simply-connected groups. This is as if one where to award a title for drinking beer, having first fixed the rules so as to exclude all citizens of Heidelberg, Munich, Burton-on-Trent, and any other place where they actually brew or drink much of the stufff. In other words, it is contrary to natural justice.
In the second place, to consider the question at all reveals a certain preoccupation wtih ordinary cohomology. Any impartial observer must marvel at your obsession with this obscure and unhelpful invariant. The author, like all respectable Lie groups, is much concerned to present a decorous and seemly appearance to the eyes of K-theory; and taken in conjunction with other general theorems, this forces him to have a modest amount of torsion in ordinary cohomology. I shall seek some suitable person to inform you in an Appendix.
As a further argument against points (1) and (2), it is natural to release some small scrap of information which you would not otherwise possess. And this may also serve to guarantee the authenticity of this letter; for you must at least believe that it comes via some mathematician who would not mislead you about my views. You may then be expecting me to reveal, for example, H*(BE8; F5). I shall not oblige you. That could only encourage you in the low tastes that I have already condemned. Instead, I shall note the following possibility. It may happen that a space Y has its K-theory K*(Y) torsion-free and zero in odd degrees, but nevertheless a careful study of K*(Y) will reveal that Y must have torsion in its ordinary cohomology. Again, I shall seek some suitable person to inform you in an Appendix.
Be it therefore known and proclaimed among you, that my K-theory K(E8) and that of my classifying space K(BE8) cannot be criticised in this respect, at least at the prime 5. Their conduct is such as would be blameless and above reproach in the K-theory of a space without 5-torsion in its ordinary cohomology.
Given at our palace, etc, etc,