Stable systolic category of the product of spheres. (English) Zbl 1214.57023

Given a closed manifold \(M\), the stable systolic category of \(M\), \(\text{cat}_{\text{stsys}}M\), is a numerical invariant (in the sense of Lusternik-Schnirelmann) which, roughly speaking, measures the volume-complexity of the manifold. The paper begins with an interesting introduction to this invariant in the broader class of local Lipschitz neighborhood retracts in a finite euclidean space.
Then, it is proven that, if \(M,N\) are manifolds of maximal real cup-length (this means in particular that their real cup-length equals their stable systolic category), then \(\text{cat}_{\text{stsys}}(M\times N)\geq \text{cat}_{\text{stsys}}M+\text{cat}_{\text{stsys}}N\).
On the other hand, without imposing maximal real cup-length, under a certain assumption on \(M\) and \(N\) involving their dimensions and the least positive degree in which they have non zero real cohomology, the author proves that \(\text{cat}_{\text{stsys}}(M\times N)= \text{cat}_{\text{stsys}}M+\text{cat}_{\text{stsys}}N\).
These results allow the author to prove the main result by which the stable systolic category of a product of spheres equals the number of the spheres involved.


57N65 Algebraic topology of manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
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[1] I K Babenko, Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) 707 · Zbl 0812.57022
[2] M Berger, Systoles et applications selon Gromov, Astérisque 216 (1993) 279 · Zbl 0789.53040
[3] A N Dranishnikov, Y B Rudyak, Stable systolic category of manifolds and the cup-length, J. Fixed Point Theory Appl. 6 (2009) 165 · Zbl 1213.55002
[4] H Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer New York, New York (1969) · Zbl 0176.00801
[5] H Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974/75) 351 · Zbl 0278.49045
[6] H Federer, W H Fleming, Normal and integral currents, Ann. of Math. \((2)\) 72 (1960) 458 · Zbl 0187.31301
[7] M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 · Zbl 0515.53037
[8] M G Katz, Y B Rudyak, Lusternik-Schnirelmann category and systolic category of low-dimensional manifolds, Comm. Pure Appl. Math. 59 (2006) 1433 · Zbl 1112.55005
[9] M G Katz, Y B Rudyak, Bounding volume by systoles of 3-manifolds, J. Lond. Math. Soc. \((2)\) 78 (2008) 407 · Zbl 1156.53024
[10] J P Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. \((2)\) 54 (1951) 425 · Zbl 0045.26003
[11] B White, Rectifiability of flat chains, Ann. of Math. \((2)\) 150 (1999) 165 · Zbl 0965.49024
[12] H Whitney, Geometric integration theory, Princeton University Press (1957) · Zbl 0083.28204
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