In the fundamental work of Postnikov [12](3) and Zilber (see the reference in [17]), one decomposes a space into a sequence or tower of fibre spaces, each of which has only a finite number of nonvanishing homotopy groups. A similar construction in the category of semisimplicial complexes was the basic technique. The procedure involved choices and was not functorial. In the semisimplicial case, J. C. Moore [10] described the construction of a natural Postnikov system for a Kan complex. In practice, his definition involves the following two difficulties: (1) To reconstruct the geometric case, one must take the singular complex of a space, perform the construction, and then apply the geometric realization functor [7]. The resulting objects are not fibre bundles. It is not clear how to describe them by means of invariants. (2) To obtain useful invariants to describe the fibre spaces in the construction, one must replace the fibre spaces by twisted Cartesian products. This process involves choices, and one no longer has a natural object.In this note, I consider Postnikov systems from a geometric point of view (see [5]). The principal result is, roughly speaking, the following: given a map of spaces f:X -* Y, there is an induced map of Postnikov systems, which sends each term in a Postnikov decomposition for X into the corresponding term in a Postnikov decomposition for Y. These individual maps are all compatible. Furthermore, the invariants (fc-invariants) for X and for Y are related via the map /. I believe that this construction will serve as an adequate substitute for a functorial Postnikov construction, which in the geometric case seems unlikely for technical reasons.There are three sections. The first is preliminary, while the second is concerned with the actual construction. In the third section, I give several applications to //-spaces, including a characterization of //-spaces with finitely many nonvanishing homotopy groups. In a subsequent note, I plan to study the group of homotopy equivalences of a space.

The group of homotopy classes of homotopy equivalences from a space X to itself, which we write as G(X), has been studies in recent years by Arkowitz and Curjel [1,2], the author [6,7] and several others. It is a natural quotient of the associative H-space made from homotopy-equivalcnces from X to itself under composition, and it has the same position, in the homotopy category, as the group of automorphisms of a group in the category of groups. It also has the practical use of determining how much the choice of k-invariants overdetermines the homotopytype of a space. G(X) is usually a complicated, non-abelian group, but we often have strong information -for example, we know in important cases, that it is finitely-presented (see [7]).Many simple questions are unfortunately not answered. For example: When does G(X) have torsion? When is G(X) finite? One basic problem, due to M.Arkowitz, asked whether there is a reasonable space X with G(X)=Z, the group of integers. The only case where we have the full picture is the fact that G(K(~, n)) = Aut (g).Our goals here are the following:1) We give a characterization of when G(X) is finite, when X is a reasonable space with finite k-invariants. Our results cover H-spaces and generalize results in [1] and [2].2) We construct a space, X, with non-trivial rational homology, for which G(X) is trivial. This shows that there is no reasonable analogue of ~-theory for G(X), even when ~ is a collection of finite groups. It also shows that one cannot expect to deduce the result of 1) above from the corresponding theorem for a product of a finite number of K(GI, 2ni + 1) spaces. We also indicate how one may get a finite complex with the same properties as X.3) We construct a reasonable space X, which has four non-vanishing homotopy groups, for which G(X)= Z.4) We prove two easy propositions which show that the situation in 2) or 3) above cannot happen in the usual stable cases.5) Finally, we indicate various open questions in this area, which should be within reach of present methods.

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