W. G. Dwyer

An arithmetic square for virtually nilpotent spaces

with E. Dror and D. M. Kan, Illinois J. Math. (21), 1977, 242-254

Math Reviews: 55:11246

Recall that a connected space is said to be nilpotent if its fundamental group is a nilpotent group and acts nilpotently on its higher homotopy groups. In spite of title of this paper, the main result it contains is the statement that any nilpotent space X fits into a Sullivan style arithmetic square. This is a homotopy fibre square which expresses X as a homotopy pullback of

  1. the product P(X) of all the p-completions of X, and
  2. the rationalization of X, over
  3. the rationalization of P(X).
This corrects an error in the original edition of Bousfield and Kan (51:1825 Bousfield, A. K.; Kan, D. M. Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972.) where it was asserted that in general such a homotopy fibre square would work only for nilpotent spaces of finite type. The p-completions in the square are the ones constructed by Bousfield and Kan, not the profinite p-completions of Sullivan (56:1305 Sullivan, Dennis Genetics of homotopy theory and the Adams conjecture. Ann. of Math. (2) 100 (1974), 1--79).

The paper also contains a somewhat technical refinement of the above statement. A connected space X is said to be virtually nilpotent if every Postnikov stage of X has a finite cover which is nilpotent. It is shown in the paper that any virtually nilpotent space is Q-good in the sense of Bousfield and Kan, as well as Z/p-good for every prime p. The question arises: what results if one forms the above arithmetic square for X and takes the homotopy pullback? Call this homotopy pullback Y. It is shown in the paper that the natural map from X to Y is a homology isomorphism. This means that Y is not for instance the Bousfield-Kan Z-completion of X, since there is an example in Bousfield-Kan of a virtually nilpotent space X (actually real projective 2-space) which is not Z-good. (Recall that X is Z-good if the map from X to the Z-completion of X is a homology isomorphism). What then is Y? It turns out that it is exactly the Bousfield localization of X with respect to integral homology (52:1676 Bousfield, A. K. The localization of spaces with respect to homology. Topology 14 (1975), 133--150).

In other words, if X is a virtually nilpotent space the arithmetic square gives a construction of the conceptually well-behaved Bousfield localization of X with respect to integral homology, in terms of computationally more accessible Bousfield-Kan completion functors.