Cohomology of Nilmanifolds, Massey products and Symplectic structures
by
Dmitri V. Millionschikov
Moscow State University, Russia

Poster

Nilmanifolds are widely used for construction of symplectic manifolds with no Kähler structure or not formal symplectic manifolds.

Let [`L]_n - a nilpotent Lie algebra with free generators e₁, e₂, ..., e_n and Lie bracket:

[ei, ej] = ì ï í ï î (j-i)ei+j, i+j <= n; 0, i+j > n.

Vector space [`L]<sub>n</sub> possesses a group structure *, defined by the Campbell-Hausdorff formula. This nilpotent group V<sub>n</sub>=([`L]_n, *) has a discrete subgroup (uniform lattice) \Gamma_n that is generated by the {e₁, ..., e_n }.

V.M.Buchstaber proposed a very nice interpretation of V_n as a special (n+1)-jet group - group of polynomial real line diffeomorphisms { t --> P_x(t)=t+x₁tⁿ⁺¹+x₂tⁿ⁺²+...+x_ntⁿ⁺¹ } with group structure defined by formula (P_x*Q_y)(t):=Q_y(P_x(t))   mod   tⁿ⁺².

Quotient spaces M_n=V_n/\Gamma_n are compact manifolds (nilmanifolds) and Eilenberg-Maclane spaces K(\Gamma_n, 1).

Rational cohomology of M_n is isomorphic to the cohomology of the Lie algebra [`L]_n (Nomizu theorem).

M_2k is symplectic manifold with the left invariant symplectic form generated by \omega_2k+1=(2k-1)\omega₁ /\ \omega_2k+(2k-3)\omega₂ /\ \omega_2k-1+...+\omega_k /\ \omega_k+1, where \omega₁, ..., \omega_2k is the dual basis to the e₁, ..., e_2k. I. K. Babenko, I. A. Taimanov used M_2k for the construction of not formal simply connected symplectic manifolds.

The problem of calculation of the cohomology of the [`L]_n is closely related to the Gontcharova's theorem in the infinite dimensional Lie algebras theory.

Namely, applying Gontcharova's theorem and Serre-Hochschild spectral sequence we get

Theorem. Stable Betti number \beta_q=dim(H^q([`L]_n)), n > 3q-1 equals to (q+2)-th Fibonacci number u_q+2.

Date received: June 1, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-05.