## Categorical sequences.(English)Zbl 1131.55001

The categorical sequence of a space $$X$$ is the function $$\sigma_X : \mathbb N \to \mathbb N \cup \{ \infty\}$$ defined by $$\sigma_X(k) =$$ inf$$\{n\,|\, \text{cat}_X(X_n) \geq k\}$$, where $$\text{cat}_X(X_n)$$ is the category of $$X_n$$ relative to $$X$$. The function $$\sigma_X$$ is a well-defined homotopy invariant of $$X$$ and the description of its properties is the object of the paper under review. A related notion is the product length sequence of a non-negatively graded commutative algebra $$A$$ defined by setting $$\sigma_A(k)$$ to be the least dimension $$n$$ for which the $$n^{th}$$ grading $$A^n$$ contains a non-trivial $$k$$-fold product. Then for any ring $$R$$, $$\sigma_X \leq \sigma_{H^*(X;R)}$$. On the other hand, if $$A$$ is the minimal model of a rational space $$X$$, then $$\sigma_X \geq \sigma_A$$. The main Theorem of the paper contains three very useful properties of $$\sigma_X$$ that make its computation quite easy and are very useful for the computation of the category of a space. The properties (b) and (c) of the Theorem are only valid in a set theory where the Whitehead problem has a positive solution ($$\text{Ext}(A,\mathbb Z) = 0$$ implies $$A$$ is free). Theorem: For any space $$X$$, (a) $$\sigma_X(k+\ell) \geq \sigma_X(k)+\sigma_X(\ell)$$, (b) if $$X$$ is simply connected and $$\sigma_X(k)=n$$, then $$H^n(X;A)\neq 0$$ for some $$A$$, (c) if equality occurs in (a) and $$X$$ is 1-connected, then the cap product $$H^k(X;A) \otimes H^{\ell}(X;B)\to H^{k+\ell}(X; A\otimes B)$$ is nontrivial for some choice of coefficients.

### MSC:

 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 55P62 Rational homotopy theory

### Keywords:

Lusternik-Schnirelmann category; rational homotopy
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