Ljusternik-Schnirelman category of skeleta
The authors study the relationship between the Ljusternik-Schnirelman category, \(cat X\), (resp. the rational LS-category, \(cat_0 X\)) of a CW-complex and the category of its \(k\)-skeleta, \(X_k\). They first prove the following theorem which is an improvement of a result due to \textit{O. Cornea} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28,No.~6, 689-704 (1995; Zbl 0858.55007)]: Theorem 1: Let \(f:X \to Y\) be a map between \((r-1)\)-connected CW complexes, where \(r \geq 1\). Assume that for some \(q \geq 0\), (i) \(\pi_i(f)\) is an isomorphism for \(i<q\) and \(\pi_q(f)\) is surjective and (ii) \(H^i(X; -)\) vanishes for \(i \geq \text{rcat } Y+q\) for all (possibly twisted) coefficients. Then \(\text{cat} X \leq \text{cat} Y\). This implies the following corollary. Corollary: If \(Y\) is a connected noncontractible CW complex then \(\text{cat } Y_k \leq \text{cat } Y\) for each \(k\)-skeleton \(Y_k\). The following result is also proved: Theorem 2: Suppose that \(Y\) is a CW-complex such that \(Y_1=\{pt\}\), \(\text{cat}_0 Y>0\), and each \(Y_k\) has finite Betti numbers. Then for sufficiently large \(N\), \[ cat_0 Y_N \leq \dots \text{cat}_0 Y_k \leq \text{cat}_0 Y_{k+1} \leq \dots \text{cat}_0 Y, \] and \[ \text{cat}_0 Y =\lim_{k \to \infty} \text{cat}_0Y_k. \]
scientific article; zbMATH DE number 1879131