Celebratio Mathematica

misc M. F. Atiyah, B. Mazur, J. W. Milnor, and V. Strassen: Addresses on the work of the 1986 Fields Medalists and Nevanlinna Prize winner, 1987. 60-min. videocassette. American Mathematical Society (Providence, RI). Plenary addresses presented at the International Congress of Mathematicians (Berkeley, CA, August 1986). MR 1064525

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incollection J. Milnor: “The work of M. H. Freedman,” pp. 405–408 in Fields Medallists’ lectures. Edited by M. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622913

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incollection M. Freedman, R. Hain, and P. Teichner: “Betti number estimates for nilpotent groups,” pp. 413–434 in Fields Medallists’ lectures. Edited by M. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622914

We prove an extension of the following result of Lubotzky and Magid on the rational cohomology of a nilpotent group \( G \): if \( b_1 < \infty \) and \( G\otimes\mathbb{Q} \neq 0 \), \( \mathbb{Q} \), \( \mathbb{Q}^2 \), then \( b_2 > b_1^2/4 \). Here the \( b_i \) are the rational Betti numbers of \( G \) and \( G\otimes\mathbb{Q} \) denotes the Malcev-completion of \( G \). In the extension, the bound is improved when we know that all relations of \( G \) all have at least a certain commutator length. As an application of the refined inequality, we show that each closed oriented 3-manifold falls into exactly one of the following classes: it is a rational homology 3-sphere, or it is a rational homology \( \mathbb{S}^1\times\mathbb{S}^2 \), or it has the rational homology of one of the oriented circle bundles over the torus (which are indexed by an Euler number \( n\in\mathbb{Z} \), e.g. \( n=0 \) corresponds to the 3-torus) or it is of general type by which we meant that the rational lower central series of the fundamental group does not stabilize. In particular, any 3-manifold group which allows a maximal torsion-free nilpotent quotient admits a rational homology isomorphism to a torsion-free nilpotent group.

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Michael H. Freedman

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