Barry Jessup

NEW BOUNDS ON THE BETTI NUMBERS OF NILPOTENT LIE ALGEBRAS

We provide new upper and lower bounds for the Betti numbers of nilpotent Lie algebras. As an application, we prove the toral rank conjecture (TRC) for nilmanifolds of dimension at most 14 and for a small class of coformal spaces. We also give a new, direct proof of the result of Deninger and Singhof that the TRC is true for 2-step nilpotent Lie algebras.

Explicit Betti numbers for a family of nilpotent Lie algebras

Betti numbers for the Heisenberg Lie algebras were calculated by Santharoubane in his 1983 paper. However few other examples have appeared in the literature. In this note we give the Betti numbers for a family of (2n+1)dimensional 2-step nilpotent extensions of R by R2n. Introduction Let g denote a finite dimensional nilpotent Lie algebra defined over an arbitrary field k. Let g∗ denote the vector space dual to g and ∧g∗ = ⊕ i≥0 ∧ig∗ the exterior algebra. The differential d : ∧g∗ → ∧g∗ is the unique derivation of degree one extending dx∗(a ∧ b) = −x∗([a, b]) for each x∗ ∈ ∧1g∗ and a, b ∈ g. We calculate the Betti numbers bi(g) of g given by bi(g) = dim(H (g, k)) for the Lie algebra cohomology with coefficients in k. For every n ∈ N, let hn denote the n Heisenberg Lie algebra. This is the (2n+1)-dimensional 2-step nilpotent Lie algebra with basis {x1, . . . , xn, y1, . . . , yn, z} and non-zero relations [xi, yi] = z for each 1 ≤ i ≤ n. According to Santharoubane [5] bi(hn) = ( 2n i ) − ( 2n i− 2 ) for all 0 ≤ i ≤ n (assuming ( p q ) = 0 unless 0 ≤ q ≤ p). The remaining numbers are given by Poincaré duality. Recall that the Heisenberg Lie algebras arise as extensions of R by R. We study a family of (2n+1)-dimensional 2-step nilpotent extensions of R by R. For every n ∈ N, let gn denote the Lie algebra with basis {x1, . . . , xn, y1, . . . , yn, z} and non-zero relations [z, xi] = yi for each 1 ≤ i ≤ n. Our main result is the following. Received by the editors April 20, 1994 and, in revised form, August 31, 1995. 1991 Mathematics Subject Classification. Primary 17B56; Secondary 17B30, 22E40.

Free Torus Actions and Two-Stage Spaces

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

Rational L-S category and conjecture of Ganea

Abstract The Lusternik-Schnirelmann category of a space satisfies cat( X × Y )≤cat( X )+cat( Y ) and the inequality can be strict. However, Ganea conjectured that equality would hold if one factor was a sphere. Halperin and Lamaire recently introduced Mcat( X ), a module-type approximation to cat( X ) satisfying Mcat( X )≤cat( X ). In some cases when F → E → B is a fibration, lower bounds for Mcat( E ) are found in terms of Mcat( B ) and invariants of F , and this proves Ganeas conjecture for Mcat.

ESTIMATING THE RATIONAL LS-CATEGORY OF ELLIPTIC SPACES.

An elliptic space is one whose rational homotopy and rational cohomol- ogy are both flnite dimensional. We prove, for Toomers invariant, two improvements of the estimate of the Mapping theorem relying on data from the homotopy Lie al- gebra of the space. In particular, we show that if S is elliptic, cat0S ‚ dim L even + dim ZL odd ; where LS is the rational homotopy Lie algebra of S and ZLS its centre. Several interesting examples are presented to illustrate our results.

On the Betti Numbers of Nilpotent Lie Algebras of Small Dimension

The work of Golod and Safarevic on class field towers motivated the conjecture that b2 > b2 1/4 for nilpotent Lie algebras of dimension at least 3, where b i denotes the i th Betti number. Using a new lower bound for b 2 and a characterization of Lie algebras of the form g/Z(g), we prove this conjecture for 2-step algebras. We also give the Betti numbers of nilpotent Lie algebras of dimension at most 7 and use them to establish the conjecture for all nilpotent Lie algebras whose centres have codimension ≤ 7.

Cohomology operations for Lie algebras

If L is a Lie algebra over R and Z its centre, the natural inclusion Z → (L*)* extends to a representation i*: ΛZ → End H*(L,R) of the exterior algebra of Z in the cohomology of L. We begin a study of this representation by examining its Poincare duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of End H* (L, R). and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to 0 → Z → L → L/Z → 0.

L-S category and homogeneous spaces

Abstract Relations between rational L-S type invariants for homogeneous spaces aregiven. If G and H are compact, simply connected Lie groups, we prove that cat 0 ( G H )≤e 0 ( G H )l 0 ( G H ) . We also show that cat 0 ( S )= e 0 ( S ) for a class of spaces which includes certain spaces of the form G (SU 2 ) n and G T k , where T is a torus. Explicit formulae for cat 0 ( S ) are given in the latter.

The rational LS-category of -trivial fibrations

We provide new upper and lower bounds for the rational LS-category of a rational fibration ξ: F → E → K(Q, 2n) of simply connected spaces that depend on a measure of the triviality of ξ which is strictly finer than the vanishing of the higher holonomy actions. In particular, we prove that if ξ is k-trivial for some k > 0 and H* (F) enjoys Poincare duality, then cato E > cat 0 F + k.

Rational Lusternik–Schnirelmann category of fibrations

Abstract If F → E → B is a fibration, a classical result of Varadarajan asserts that cat E⩽ cat F+ cat B( cat F+1) , where cat S denotes the Lusternik–Schnirelmann category of S . We give improved upper bounds in the rational case of the form cat 0 E⩽ cat 0 F+ cat 0 B( cat 0 F+2−r 0 F), where r 0 F is a new invariant, namely the rational retraction index of F satisfying depth F⩽r 0 F⩽ cat 0 F, so that we recover the classical formula when r 0 F =1. However, the retraction index is often larger than 1, and in particular, we prove that if H ∗ (F; Q ) is a Poincare duality algebra with at least 2 generators, then r 0 F ⩾2, giving the bound of (Contemp. Math. 227 (1996) 177) without their dimension hypothesis. Moreover, if F is coformal, then r 0 F= cat 0 F , which yields the much lower estimate cat 0 E⩽ cat 0 F+2 cat 0 B.