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From Combinatorics to Topology Via Algebraic Isoperimetry

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From Combinatorics to Topology Via Algebraic Isoperimetry Geom. Funct. Anal. Vol. 20 (2010) 416–526 DOI 10.1007/s00039-010-0073-8 Published online July 30, 2010 © 2010 Springer Basel AG GAFA Geometric And Functional Analysis SINGULARITIES, EXPANDERS AND TOPOLOGY OF MAPS. PART 2: FROM COMBINATORICS TO TOPOLOGY VIA ALGEBRAIC ISOPERIMETRY Mikhail Gromov Abstract. We find lower bounds on the topology of the fibers F −1(y) ⊂ X of continuous maps F : X → Y in terms of combinatorial invariants of certain polyhedra and/or of the cohomology algebras H∗(X). Our exposition is conceptually related to but essentially independent of Part 1 of the paper. Contents 1 Definitions, Problems and Selected Inequalities 417 1.1 Δ-Inequalities for the multiplicities of maps of the n-skeleton of the N-simplex to Rn 418 1.2 (n − k)-Planes crossing many k-simplices in Rn ..................... 419 1.3 Riemannian and sub-Riemannian waist inequalities . 420 1.4 Locally bounded 2-polyhedra with large cardinalities over R2 ............. 422 1.5 Separation inequalities in the N-torus for the homological μA-mass.......... 423 1.6 |A|F-Isoperimetryandhomologyexpanders....................... 425 2 Filling Profiles, Random Cones in Spaces of Cycles and Lower Bounds on Mul- tiplicities of Maps 426 2.1 Semisimplicial spaces and S-construction........................ 426 n 2.2 Spaces clsms of n-cycles in chain complexes, quasitransversality and the intersection ∗ homomorphism F ◦ n .................................. 429 −1 2.3 Filling norms ∂ fil in metric and measurable chain complexes . 431 n 2.4 Compounded filling profiles Φi of metric complexes and contractions in spaces of cycles 433 2.5 Lower bounds on multiplicities of maps of polyhedra to manifolds . 435 −1 2.6 Random cones, bound ∂ rand ≤ 1inΔ(V ) and the proof of the topological Δ- inequality.......................................... 436 2.7 Crofton–Wendel formula, waist inequalities and relative growth of infinite groups . 439 2.8 Randomized Radon–Tverberg theorem . 443 2.9Openquestionsonmultiplepointsofmaps........................ 445 2 2.10 l2-Filling bounds and maps of 2-polyhedra to R ..................... 448 2.11Isoperimetryincubes.................................... 450 2.12Fillingwithsubdividedcones................................ 454 2.13Coloredpolyhedraandsphericalbuildings......................... 456 2.14Randompolyhedra...................................... 458 Keywords and phrases: Expanders, isoperimetric inequalities, semisimplicial complexes, spaces of cycles 2010 Mathematics Subject Classification: 05C10, 05C80, 06F05, 53C23, 05D05, 55U10. GAFA COMBINATORICS TO TOPOLOGY VIA ALGEBRAIC ISOPERIMETRY 417 3 Appendices to Section 2 461 3.1 Isoperimetry with ∂-boundary and cardinality of graphs over graphs . 461 3.2 Barany–Boros–Furedi inequality for cliques in the plane . 465 3.3 Filling-in geometric cycles via the local-to-global variational principle. 466 3.4Moves,tubesandfillinginequalities........................... 472 3.5 Lower bounds on volumes of minimal varieties and waist inequalities. 476 −1 3.6 Crofton’s formulas and lower bounds on ∂ fil bycalibrations............ 478 1 −1 3.7 Bound (∂ ) fil(β) < 1 for small β ........................... 482 3.8 Bounds on ∂i andTuran’sgraphs........................... 483 4 Homological Isoperimetry 4.1 Ideal valued measures and cohomological widths . 486 4.2Maximalfiberinequalityrevisited............................. 489 4.3Constructionofsimplyconnectedexpanders....................... 491 4.4Coarsegeometryversustopology............................. 493 4.5 Separation and isoperimetric profiles of graded semigroups and algebras . 498 4.6 Combinatorial reduction of separation inequalities in orderable monomial algebras . 502 4.7 ∂-isoperimetryingraphalgebras............................ 503 4.8 ∂-control of homological isoperimetry and topological translation of extremal set systemsinequalities..................................... 506 4.9 Parametric separation inequalities . 509 4.10 Homological realization of monomial algebras by spaces with locally bounded geome- tries............................................. 510 4.11 Maps to trees and folds in R2 ............................... 514 4.12Tree-likealgebrasovergraphs............................... 516 4.13Perspectivesandproblems................................. 519 1 Definitions, Problems and Selected Inequalities The simplest measure of the fiber-wise complexity of a continuous map between equidimensional topological spaces, F : X → Y , is the maximal cardinality of a | −1 | fiber, supy∈Y F (y) . The problem we address is that of evaluating the cardinality of X over Y ,definedas inf sup F −1(y) for a given class F of maps F : X → Y. F ∈F y∈Y When X is a polyhedron, it is better to deal with the maximal number of closed cells in X intersected by a fiber, since the latter may only increase under uniform limits of maps. Next, we extend the setting to maps from k-dimensional X to (k+l)-dimensional Y with a distinguished family of l-dimensional subspaces A ⊂ Y wherewesearch for an A0 which is intersected by a maximal number of k-cells of X mapped to Y . Finally, we establish lower bounds on the (co)homologies of fibers of continuous maps rather than on their cardinalities. Acknowledgments. The final version of this paper owes very much to the anony- mous referee who, besides indicating a multitude of minor errors, pointed out several significant mistakes in the original draft and who has made a variety of useful sug- gestions. Also I want to thank Miriam Hercberg for her help in proofreading the paper. 418 M. GROMOV GAFA 1.1 Δ-Inequalities for the multiplicities of maps of the n-skeleton of the N-simplex to Rn. Let X be a finite simplicial polyhedron, where the set of the k-faces (simplices) Δk in X is denoted by {Δk} = {Δk}(X)and|{Δk}| denotes the cardinality of this set. Let F : X → Rn be a continuous (e.g. face-wise affine) map and let A be an Rn affine subspace in . Denote by k A F {Δ } the number of the closed k-faces Δk of X such that the image F (Δk) ⊂ Rn inter- sect A. We want to identify and/or construct polyhedra X of “sufficient combinatorial complexity”, for which every continuous (or, at least, every face-wise affine) map F from X Rn − n−k ⊂ Rn to admits an (n k)-dimensional affine subspace A0 = A0 (which depends on F ), such that the ratio k |A0 F {Δ }| |{Δk}| is “reasonably large”. A particular case of interest is where k = n,whereA = a are points in Rn and where the first result (I am aware of) of this kind (apart from Tverberg’s theorem, N see2.8),duetoImreBarany[Ba],concernsaffine maps of the N-simplex X =Δ Rn |{ n}| N+1 to ,where,observe,thenumberofthen-faces is Δ = n+1 . Barany’s affine Δ-inequality. N n Let F :Δ → R be an affine map. Then there exists a point a0 = n n a0(F ) ∈ R such that the number M0 of the closed n-faces Δ of the N n n simplex Δ for which the image F (Δ ) ⊂ R contains a0 is bounded from below by n n M0 = a0 F {Δ } ≥ baff (n) · {Δ } · 1 − O(1/N ) , for n+1 n baff (n) ≥ 1/(n +1) . [Δ → R ]aff Equivalently, n n given a probability measure μ on R , there exists a point a0 ∈ R such that the convex hull of μ-randomly chosen n +1 points in Rn contains n+1 a0 with probability ≥ 1/(n +1) . The true value of baff is known only for n = 1, where it is, obviously, 1/2, and for n = 2, where the lower bound baff (2) ≥ 2/9isduetoBorosandFuredi[BorF], while the examples constructed in [BuMN] show that baff (2) ≤ 2/9; moreover it is −n shownin[BuMN]thatbaff (n) e for large n. n+1 Barany’s bound baff (n) ≥ 1/(n +1) was improved by a polynomial factor n in [W] (also see [DeHST]), but an exponential lower bound, baff (n) ≥ β for some β>0, remains problematic. (I am indebted to Janoch Pach who introduced me to the combinatorial results around the Barany inequality.) GAFA COMBINATORICS TO TOPOLOGY VIA ALGEBRAIC ISOPERIMETRY 419 Topological Δ-inequality. Let F :ΔN → Rn be a continuous map. Then there exists a point n n a0 ∈ R , such that the number M0 of the closed n-faces Δ of the simplex N n n Δ for which the image F (Δ ) ⊂ R contains a0 is bounded from below by n n M0 = a0 F {Δ } ≥ btop(n) · {Δ } · 1 − O(1/N ) , for 2n b (n) ≥ ∼ en/(n +1)n+1. [Δ → Rn] top (n +1)(n +1)! top (Strangely, this is stronger than the present day bound on baff obtained by the traditional combinatorial techniques. A priori, the constant btop which serves all continuous maps may be only smaller than the constant baff responsible only for the face-wise affine maps. The two constants happen to be equal for n =1, 2, but it is unknown if btop(n) <baff (n) for any n ≥ 3.) The proof of the topological Δ-inequality depends on “combinatorial filling” in the semisimplicial spaces of cycles of measurable chain complexes (see 2.1–2.6) which is similar to, but formally independent of, the filling techniques in the Riemannian geometry. Interestingly enough this quite formal “combinatorial filling” yields lower bounds on the waists of Riemannian and (some sub-Riemannian) spaces under less restrictive assumptions than those required by the known geometric arguments (see 1.3, 2.7). The (more or less standard) geometric ideas motivating our general constructions are presented in section 3 which includes: 2 • A separate proof of the inequality [Δ → R ]aff , which contains in a nutshell the idea of our general argument; • Basic definitions of the Riemannian geometry and a brief overview of the isoperimetric/filling techniques aimed at non-experts; • Basics on isoperimetry in graphs and cardinalities of graphs over graphs. n 1.2 (n − k)-Planes crossing many k-simplices in R . Denote by baff (n, k), where n ≥ k, the maximal number, such that every probability measure μ in Rn, admits an (n − k)-dimensional affine n−k subspace A0 = A0 (μ) such that the μ-probability of the convex hull of Rn k +1 points in to meet A0 is bounded from below by ⊗(k+1) k μ A0 {Δ } ≥ baff (n, k) .
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