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

Ryu, Hoil

Algebr. Geom. Topol. 11, No. 2, 983-999 (2011).

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.

Reviewer: Aniceto Murillo (Malaga)

MSC:

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)

Keywords:

stable systolic cateogry; real cup-length; homology systole

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