**Mathematical Surveys and Monographs**

Volume: 137;
2007;
222 pp;
Hardcover

MSC: Primary 53;
Secondary 11; 16; 17; 28; 30; 37; 52; 55; 57

Print ISBN: 978-0-8218-4177-8

Product Code: SURV/137

List Price: $75.00

Individual Member Price: $60.00

**Electronic ISBN: 978-1-4704-1364-4
Product Code: SURV/137.E**

List Price: $75.00

Individual Member Price: $60.00

#### Supplemental Materials

# Systolic Geometry and Topology

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*Mikhail G. Katz*

The systole of a compact metric space \(X\) is a metric invariant of \(X\), defined as the least length of a noncontractible loop in \(X\). When \(X\) is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by W. Tutte. The first nontrivial results for systoles of surfaces are the two classical inequalities of C. Loewner and P. Pu, relying on integral-geometric identities, in the case of the two-dimensional torus and real projective plane, respectively. Currently, systolic geometry is a rapidly developing field, which studies systolic invariants in their relation to other geometric invariants of a manifold.

This book presents the systolic geometry of manifolds and polyhedra, starting with the two classical inequalities, and then proceeding to recent results, including a proof of M. Gromov's filling area conjecture in a hyperelliptic setting. It then presents Gromov's inequalities and their generalisations, as well as asymptotic phenomena for systoles of surfaces of large genus, revealing a link both to ergodic theory and to properties of congruence subgroups of arithmetic groups. The author includes results on the systolic manifestations of Massey products, as well as of the classical Lusternik-Schnirelmann category.

#### Table of Contents

# Table of Contents

## Systolic Geometry and Topology

- Contents vii6 free
- Preface xi10 free
- Acknowledgments xiii12 free
- Part 1. Systolic geometry in dimension 2 114 free
- Chapter 1. Geometry and topology of systoles 316
- Chapter 2. Historical remarks 1326
- Chapter 3. The theorema egregium of Gauss 2134
- Chapter 4. Global geometry of surfaces 2942
- Chapter 5. Inequalities of Loewner and Pu 3952
- Chapter 6. Systolic applications of integral geometry 4356
- 6.1. An integral-geometric identity 4356
- 6.2. Two proofs of the Loewner inequality 4457
- 6.3. Hopf fibration and the Hamilton quaternions 4659
- 6.4. Double fibration of SO(3) and integral geometry on S[sup(2)] 4659
- 6.5. Proof of Pu's inequality 4861
- 6.6. A table of optimal systolic ratios of surfaces 4861

- Chapter 7. A primer on surfaces 5164
- Chapter 8. Filling area theorem for hyperelliptic surfaces 5770
- 8.1. To fill a circle: an introduction 5770
- 8.2. Relative Pu's way 5972
- 8.3. Outline of proof of optimal displacement bound 6073
- 8.4. Near optimal surfaces and the football 6174
- 8.5. Finding a short figure eight geodesic 6376
- 8.6. Proof of circle filling: Step 1 6376
- 8.7. Proof of circle filling: Step 2 6477

- Chapter 9. Hyperelliptic surfaces are Loewner 6982
- Chapter 10. An optimal inequality for CAT(0) metrics 7588
- Chapter 11. Volume entropy and asymptotic upper bounds 8598

- Part 2. Systolic geometry and topology in n dimensions 91104
- Chapter 12. Systoles and their category 93106
- 12.1. Systoles 93106
- 12.2. Gromov's spectacular inequality for the 1-systole 95108
- 12.3. Systolic category 97110
- 12.4. Some examples and questions 99112
- 12.5. Essentialness and Lusternik-Schnirelmann category 100113
- 12.6. Inessential manifolds and pullback metrics 101114
- 12.7. Manifolds of dimension 3 102115
- 12.8. Category of simply connected manifolds 104117

- Chapter 13. Gromov's optimal stable systolic inequality for CP[sup(n)] 107120
- Chapter 14. Systolic inequalities dependent on Massey products 113126
- Chapter 15. Cup products and stable systoles 119132
- 15.1. Introduction 119132
- 15.2. Statement of main results 120133
- 15.3. Results for the conformal systole 122135
- 15.4. Some topological preliminaries 124137
- 15.5. Ring structure-dependent bound via Banaszczyk 125138
- 15.6. Inequalities based on cap products, Poincaré duality 127140
- 15.7. A sharp inequality in codimension 1 129142
- 15.8. A conformally invariant inequality in middle dimension 130143
- 15.9. A pair of conformal systoles 130143
- 15.10. A sublinear estimate for a single systole 133146

- Chapter 16. Dual-critical lattices and systoles 135148
- 16.1. Introduction 135148
- 16.2. Statement of main theorems 135148
- 16.3. Norms on (co-) homology 137150
- 16.4. Definition of conformal systoles 138151
- 16.5. Jacobi variety and Abel-Jacobi map 139152
- 16.6. Summary of the proofs 140153
- 16.7. Harmonic one-forms of constant norm and flat tori 141154
- 16.8. Norm duality and the cup product 144157
- 16.9. Hölder inequality in cohomology and case of equality 146159
- 16.10. Proof of optimal ( 1, n – 1)-inequality 147160
- 16.11. Consequences of equality, criterion of dual-perfection 148161
- 16.12. Characterisation of equality in ( 1, n – 1)-inequality 149162
- 16.13. Construction of all extremal metrics 151164
- 16.14. Submersions onto tori 152165

- Chapter 17. Generalized degree and Loewner-type inequalities 155168
- 17.1. Burago-Ivanov-Gromov inequality 155168
- 17.2. Generalized degree and BIG(n, b) inequality 156169
- 17.3. Pu's inequality and generalisations 157170
- 17.4. A Pu times Loewner inequality 158171
- 17.5. A decomposition of the John ellipsoid 159172
- 17.6. An area-nonexpanding map 159172
- 17.7. Proof of BIG( n, b)-inequality and Theorem 17.4.1 161174

- Chapter 18. Higher inequalities of Loewner-Gromov type 163176
- 18.1. Introduction, conjectures, and some results 163176
- 18.2. Notion of degree when dimension exceeds Betti number 164177
- 18.3. Conformal BIG(n, p)-inequality 166179
- 18.4. Stable norms and conformal norms 168181
- 18.5. Existence of L[sup(p)]-minimizers in cohomology classes 169182
- 18.6. Existence of harmonic forms with constant norm 171184
- 18.7. The BI construction adapted to conformal norms 173186
- 18.8. Abel-Jacobi map for conformal norms 174187
- 18.9. Attaining the conformal BIG bound 174187

- Chapter 19. Systolic inequalities for L[sup(p)] norms 177190
- Chapter 20. Four-manifold systole asymptotics 181194
- 20.1. Schottky problem and the surjectivity conjecture 181194
- 20.2. Conway-Thompson lattices CT[sub(n)] and idea of proof 183196
- 20.3. Norms in cohomology 183196
- 20.4. Conformal length and systolic flavors 184197
- 20.5. Systoles of definite intersection forms 185198
- 20.6. Buser-Sarnak theorem 186199
- 20.7. Sign reversal procedure SR and Aut(I[sub(n,1)])-invariance 186199
- 20.8. Lorentz construction of Leech lattice and line CT[sup(⊥)][sub(n)] 187200
- 20.9. Three quadratic forms in the plane 189202
- 20.10. Replacing λ[sub(1)]i by the geometric mean (λ[sub(1)]λ[sub(2)])[sup(1/2)] 190203
- 20.11. Period map and proof of main theorem 192205

- Appendix A. Period map image density (by Jake Solomon) 195208
- A.1. Introduction and outline of proof 195208
- A.2. Symplectic forms and the self-dual line 196209
- A.3. A lemma from hyperbolic geometry 197210
- A.4. Diffeomorphism group of blow-up of projective plane 198211
- A.5. Background material from symplectic geometry 199212
- A.6. Proof of density of image of period map 201214

- Appendix B. Open problems 205218
- Bibliography 209222
- Index 221234

#### Readership

Graduate students and research mathematicians interested in new methods in differential geometry and topology.