Title of Paper: On the realization of the unstable modules

Author: JIANG Dong Hua

AMS Classification numbers: 55N99, 55S10

math.AT/0212054

Address of Author: LAGA, Institut Galilee, UMR 7539
University Paris Nord, Avenue Jean-Baptiste Clement
93430 VILLETANEUSE, FRANCE

Email address of Author: donghua.jiang@m4x.org

In this article, we give some restrictions about the structure of an
unstable module, which should be verified providing this module is the
reduced mod 2 cohomology of a space or a spectrum. We begin by studing
the structure of the sub-modules of \Sigma^s H^\ast(B(Z/2)^{\oplus d};
Z/2)^{\oplus \alpha_d} (s \geq 0, \alpha_d > 0), i.e., the unstable
modules whose nilpotent filtration has length 1. Next, we generelise
this result for the unstable modules whose nilpotent filtration has a
finite length, and who verified an additional condition. The result says
that under some hypothesis, the reduced mod 2 cohomology of a space or a
spectrum does not have arbitrary big gaps in its structure. This result
is obtained by applying the famous Adams' theorem about the Hopf
invariant and the classification of the injective unstable modules.

For the unstable modules satisfing the condition of the theorem 3 (for
example, any suspension of a sub-module of H^\ast(B(Z/2)^{\oplus d};
Z/2)^{\oplus \alpha_d}, the theorem 3 gives the upper bound of the
length of the gaps in the modules, which means the module does not
contain arbitrary big gaps. So when the module is reduced satisfing the
condition of the theorem 4, its weight should be infinite. This gives us
so many examples of the non-realizable unstable modules: F(n), any
tensor product of F(n_i), etc. (These examples can also be proved by the
theorem of Lionel Schwartz about the Kuhn conjecture, which was
generalised by F-X. Dehon - G. Gaudens.)

This article is written in french and the work is done under the
direction of L. Schwartz.