Manfred Hartl, 
Structures polynomiales en th\'eorie des groupes nilpotents. 

Abstract : As an approximation of group cohomology $H^*(G,M)$ with 
coefficients in a nilpotent module, a theory of polynomial cohomology 
$P_nH^*(G,M)$ of degre $n\ge 0$ is introduced, by passing to a functorial 
quotient of the bar resolution which is of finite type over ${\bf Z}$ in 
all dimensions if $G$ is of finite type. Thus polynomial cohomology is well
computable, and its elements are canonically represented by polynomial 
(indeed, numerical) cocycles. This construction generalizes a concept of 
Passi's. It is proved that $P_nH^2(G,M) = H^2(G,M)$ if $G$ is a torsionfree 
nilpotent group of finite type (= T-group) and $M$ is an additively 
torsionfree nilpotent $G$-module, as soon as $n+1 \ge$ the sum of the 
nilpotency classes of $G$ and $M$. This allows for an effective inductive 
construction of T-groups in terms of polynomial group laws on free abelian 
groups. The homology group $H_2(G)$ is computed for all 2-step nilpotent 
groups; an easily evaluable formula is given for the finite case. Based on 
the construction of ``abelian models" for extensions of T-groups, the 
automorphism groups of T-groups are determined in terms of iterated 
extensions. All associated obstruction operators and extension classes are 
exhibited, thus revealing unexpected simple structures. Concerning a 
question of Milnor's, we construct (with K. Dekimpe) complete left-symmetric 
(and so affine) structures on a large class of 4-step nilpotent Lie algebras, 
by studying certain submodules of polynomial constructions on Lie algebras. 
This provides a realization of certain 4-step nilpotent T-groups as 
fundamental groups of compact, complete affinely flat manifolds. A final 
chapter is devoted to subgroups induced by ideals of group algebras. A 
universal coefficient composition is obtained for such subgroups, 
generalizing the one of Sandling, Passi, Parmenter and Sehgal's for Lie and 
ordinary dimension subgroups. Relative versions of Fox and dimension 
subgroups are introduced and determined in low dimensions. The problem of 
constructing new counterexamples to the classical dimension subgroups 
conjecture is studied in dimension 4, combining Passi's homological 
viewpoint with our results on the Schur multiplier of 2-step nilpotent 
groups. Thereby a systematical systematical construction of families of 
such examples with prescribed properties is obtained.