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Product Code:  SURV/137 
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Hardcover ISBN:  9780821841778 
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Hardcover ISBN:  9780821841778 
Product Code:  SURV/137 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413644 
Product Code:  SURV/137.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821841778 
eBook ISBN:  9781470413644 
Product Code:  SURV/137.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 137; 2007; 222 ppMSC: Primary 53; Secondary 11; 16; 17; 28; 30; 37; 52; 55; 57
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 integralgeometric identities, in the case of the twodimensional 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 LusternikSchnirelmann category.
ReadershipGraduate students and research mathematicians interested in new methods in differential geometry and topology.

Table of Contents

Chapters

1. Geometry and topology of systoles

2. Historical remarks

3. The theorema egregium of Gauss

4. Global geometry of surfaces

5. Inequalities of Loewner and Pu

6. Systolic applications of integral geometry

7. A primer on surfaces

8. Filling area theorem for hyperelliptic surfaces

9. Hyperelliptic surfaces are Loewner

10. An optimal inequality for CAT(0) metrics

11. Volume entropy and asymptotic upper bounds

12. Systoles and their category

13. Gromov’s optimal stable systolic inequality for $CP^n$

14. Systolic inequalities dependent on Massey products

15. Cup products and stable systoles

16. Dualcritical lattices and systoles

17. Generalized degree and Loewnertype inequalities

18. Higher inequalities of LoewnerGromov type

19. Systolic inequalities for $L^p$ norms

20. Fourmanifold systole asymptotics


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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 integralgeometric identities, in the case of the twodimensional 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 LusternikSchnirelmann category.
Graduate students and research mathematicians interested in new methods in differential geometry and topology.

Chapters

1. Geometry and topology of systoles

2. Historical remarks

3. The theorema egregium of Gauss

4. Global geometry of surfaces

5. Inequalities of Loewner and Pu

6. Systolic applications of integral geometry

7. A primer on surfaces

8. Filling area theorem for hyperelliptic surfaces

9. Hyperelliptic surfaces are Loewner

10. An optimal inequality for CAT(0) metrics

11. Volume entropy and asymptotic upper bounds

12. Systoles and their category

13. Gromov’s optimal stable systolic inequality for $CP^n$

14. Systolic inequalities dependent on Massey products

15. Cup products and stable systoles

16. Dualcritical lattices and systoles

17. Generalized degree and Loewnertype inequalities

18. Higher inequalities of LoewnerGromov type

19. Systolic inequalities for $L^p$ norms

20. Fourmanifold systole asymptotics