Poincaré polynomials of a map and a relative Hilali conjecture (Q2143687)

Let \(X\) be a simply-connected rational elliptic space. By definition, the space \(X\) satisfies the condition that \(\dim H_*(X; {\mathbb Q}) := \sum_{i\geqq 0}\dim H_i(X; {\mathbb Q})\) and \(\dim \pi_*(X)\otimes{\mathbb Q} := \sum_{i\geqq 2} \pi_i(X)\otimes{\mathbb Q}\) are finite. The Hilali conjecture [\textit{M. R. Hilali}, Action du tore \(T^n\) sur les espaces simplement connexes. Université catholique de Louvain (PhD thesis) (1980)] claims that for such a space \(X\) the inequality \[ \dim \pi_*(X)\otimes{\mathbb Q} \leqq \dim H_*(X; {\mathbb Q}) \] holds. So far, no counterexample to the conjecture has been found. In [Tbil. Math. J. 12, No. 4, 123--129 (2019; Zbl 1436.55019)], the second named author proves that the Hilali conjecture always holds ``modulo product''; that is, there exists a positive number \(n_0\) such that \(\dim \pi_*(X^n)\otimes{\mathbb Q} < \dim H_*(X^n; {\mathbb Q})\) for \(n \geqq n_0\), where \(X^n\) denotes the \(n\)-fold product of \(X\). The paper considers a \textit{relative} version of the conjecture modulo product. We say that a continuous map \(f : X\to Y\) between simply-connected spaces is \textit{rationally elliptic with respect to kernel} if \(\sum_{i}\text{Ker} ( \dim H_i(f; {\mathbb Q})) < \infty\) and \(\sum_{i}\text{Ker}(\dim \pi_i(f)\otimes{\mathbb Q}))<\infty\). Then the authors define polynomials \(P_f(t)\) and \(P^\pi_f(t)\) by \[ P_f(t) := 1 + \sum_{i\geq 2}t^i \ \text{Ker}(\dim H_i(f; {\mathbb Q})) \ \ \text{and} \ \ P^\pi_f(t):= \sum_{i\geqq 2}t^i \ \text{Ker}(\dim \pi_*(f)\otimes{\mathbb Q}), \] respectively. The main theorem of the paper is as follows. \textbf{Theorem 3.8.} Let \(f : X\to Y\) be a continuous rationally elliptic map with respect to kernel of simply-connected spaces \(X\) and \(Y\) such that the homology rank of the source \(X\) is finite. If \(P_f(1)>1\), i.e., there exists some integer \(i\) such that \(H_i(f; {\mathbb Q}) : H_i(X; {\mathbb Q}) \to H_i(Y; {\mathbb Q})\) is not injective, then there exists some integer \(n_0\) such that for \(n \geqq n_0\) the strict inequality \(P^\pi_{f^n}(1) < P_{f^n}(1)\) holds, i.e., \[ \sum_{i\geqq 2}\text{Ker}(\dim \pi_i(f^n)\otimes{\mathbb Q}) < 1 + \sum_{i\geqq 2}\text{Ker}(\dim H_i(f^n; {\mathbb Q})). \] This is regarded as a \textit{relative Hilali conjecture ``modulo product''}. Observe that if \(Y\) is contractible, then Theorem 3.8 indeed implies the result on the Hilali conjecture modulo product mentioned above. The keys to proving the theorem are multiplicativity formulae for polynomials \(P^\pi_f(t)\) and \(P_f(t)\) which assert that \(P^\pi_{f_1\times f_2}(t)= P^\pi_{f_1}(t)+ P^\pi_{f_2}(t)\) for each \(t\) and \(P_{f_1}(t) \times P_{f_2}(t) \leqq P_{f_1\times f_2}(t)\) for \(t \geqq 0\). Moreover, an elementary fact in calculus that \(\displaystyle{\lim_{n\to \infty}}nr^n =0\) if \(|r| < 1\) is used in the proof. The paper is concluded after discussing cases of rational hyperbolic spaces. In particular, the authors pose the following question. \textbf{Question 4.3.} (a Hilali conjecture in the hyperbolic case) Let \(X\) be a rational hyperbolic space and \(P_X(t)\) denote the homological Hilbert-Poincaré series of \(X\). Let \(r_X\) be the radius of convergence of the homotopical Hilbert-Poincaré series \(HP^\pi_X(t):= \sum_{i\geqq 2}^\infty (\dim \pi_i(X)\otimes{\mathbb Q}) \ t^i\). Suppose that \(HP^\pi_X(t)\) converges at \(r_X\). Does the inequality \(HP^\pi_X(r_X) \leqq P_X(r_X)\) hold?