The categorical sequence of a rational space (Q1625447)

The categorical sequence of a space \(X\), initially introduced and studied in [\textit{R. Nendorf} et al., Algebr. Geom. Topol. 6, 809--838 (2006; Zbl 1131.55001)] is a sequence \(\sigma _X : \mathbb{N}\rightarrow \mathbb{N}\cup \{\infty\}\) of integers defined in terms of the relative Lusternik-Schnirelmann categories \(cat_X(X_n)\) of its skeleta by setting: \[ \sigma _X(k) = \text{inf}\{n \mid cat_X(X_n) \geq k\}. \] It is a homotopy invariant satisfying the following properties: \[ \sigma _X(k)=n\Rightarrow cat_X(X_{n-1})\leq k-1 < k \leq cat_X(X_n). \quad (1) \] \[ \sigma _X(k_1+k_2+\ldots +k_r)\geq \sigma _X(k_1)+\sigma _X(k_2)+\ldots +\sigma _X(k_r);\; k_1, k_2, \ldots , k_r\in \mathbb{N}.\quad (2) \] If \(X\) {is simply-connected and equality holds in (2) for some} \(k_1, \ldots , k_r\in \mathbb{N}\), {then the exterior cap product map} \[ \tilde{H}^{n_1}(X;G_1)\otimes \tilde{H}^{n_2}(X;G_2)\otimes \cdots \otimes\tilde{H}^{n_r}(X;G_r)\rightarrow \tilde{H}^{n_1+\cdots +n_r}(X;\otimes _{i=1}^{i=r} G_i)\quad (3) \] \(\text{is nonzero for some choices of coefficient groups} \; G_1,\; G_2,\; \ldots ,G_r.\) This paper studies initial three-term segments of categorical sequences of finite type rational spaces. The authors show in fact that for sequences of the form \(u=(a, b, c, \ldots )\) with \(a\geq 2\) and \(b\geq 2a\): {\parindent=6mm \begin{itemize}\item[(i)] (Theorem 1.1) If \(c=a+b\), then there exists a rational space \(X\) of finite type such that \(\sigma_X = u\) if and only if \(b\equiv 2\) mod \(a-1\). \item[(ii)] (Theorem 1.2) If \(c= s(a-1)+ta+2\) for some integers \(s, t>0\) and \(2a\leq b < c-a\), then there is a simply connected rational space \(X\) of finite type such that \(\sigma_X = u\). \end{itemize}} The proof of (i) is made in three steps. In \S 3, the authors first reduce their search for rational spaces \(X\) satisfying \(\sigma_X=(a, b, a+b, \ldots )\) to Postnikov sections of wedges of rational spheres (Theorem 3.1). Afterwards, in \S 5.2, they prove that, without the condition \(b\equiv 2\) mod \(a-1\), no sequence \(u=(a, b, a+b, \ldots )\) can be realized as a categorical sequence of a finite type rational space. In \S 6, they make use of the Leray-Serre spectral sequence associated to the fibration \(W(b-1)\rightarrow W\rightarrow P\) where \(P=P_{b-1}(W)\) denotes the \((b-1)^{th}\) Postnikov section of a finite wedge of rational spheres \(W\), to establish constraints on the first three terms of the categorical sequence \(\sigma_P\). In \S 6.3, a specific \(W\) whose \(P_{b-1}(W)\) realizes the sequence \(u=(a, b, a+b, \ldots)\) is constructed. For the assertion (ii), thanks to Lemma 6.1, the authors obtain an essential reduction to the case where \(b\not\equiv 1\) mod \(a - 1\). They complete then the realization of \(u=(a, b, c, \ldots )\) by adapting the method used in the preceding case.