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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
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