Systolic inequalities and Massey products in simply-connected manifolds.pdf

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Systolic inequalities and Massey products in simply-connected manifolds.pdf

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Systolic inequalities and Massey products in simply-connected manifolds

a r X i v : m a t h / 0 6 0 4 0 1 2 v 2 [ m a t h .D G ] 9 N o v 2 0 0 6 SYSTOLIC INEQUALITIES AND MASSEY PRODUCTS IN SIMPLY-CONNECTED MANIFOLDS MIKHAIL KATZ? Abstract. We show that the existence of a nontrivial Massey product in the cohomology ring H?(X) imposes global constraints upon the Riemannian geometry of a manifold X . Namely, we exhibit a suitable systolic inequality, associated to such a prod- uct. This generalizes an inequality proved in collaboration with Y. Rudyak, in the case when X has unit Betti numbers, and real- izes the next step in M. Gromov’s program for obtaining geometric inequalities associated with nontrivial Massey products. The in- equality is a volume lower bound, and depends on the metric via a suitable isoperimetric quotient. The proof relies upon W. Ba- naszczyk’s upper bound for the successive minima of a pair of dual lattices. Such an upper bound is applied to the integral lattices in homology and cohomology of X . The possibility of applying such upper bounds to obtain volume lower bounds was first exploited in joint work with V. Bangert. The latter work deduced systolic inequalities from nontrivial cup-product relations, whose role here is played by Massey products. Contents 1. Volume bounds and systolic category 2 2. Massey products and isoperimetric quotients 3 3. The results 4 4. Banaszczyk’s bound for the successive minima of a lattice 6 5. Linearity vs. indeterminacy of triple Massey products 7 6. Proofs of main results 10 7. Acknowledgments 12 References 12 1991 Mathematics Subject Classification. Primary 53C23; Secondary 55S30, 57N65 . Key words and phrases. isoperimetric quotient, Lusternik-Schnirelmann cate- gory, Massey product, quasiorthogonal bases, successive minima, systole. ?Supported by the Israel Science Foundation (grants no. 620/00, 84/03, and 1294/06). 1 2 M. KATZ 1. Volume bounds and systolic category A general framework for systolic geometry in a topological context was proposed in [KR06], in terms of