论文标题
地图和相对希拉利猜想的庞加莱多项式
Poincaré polynomials of a map and a relative Hilali conjecture
论文作者
论文摘要
In this paper we introduce homological and homotopical Poincaré polynomials $P_f(t)$ and $P^π_f(t)$ of a continuous map $f:X \to Y$ such that if $f:X \to Y$ is a constant map, or more generally, if $Y$ is contractible, then these Poincaré polynomials are respectively equal to the usual homological and homotopical Poincaré多项式$ p_x（t）$和$ p^π_x（t）$ $ x $。我们的相对希拉利猜想$ p^π_f（1）\ leqq p_f（1）$是众所周知的希拉利猜测$ p^π_x（1）\ leqq p_x（1）$的地图版本。 \ to h_i（y; \ Mathbb Q）$对某些$ i> 0 $不可注入，地图的相对hilali猜想所持的，即存在一个积极的整数$ n_0 $，以至于$ \ forall n \ forall n \ forall n \ geqq n_0 $ geqq n_0 $ p_ {f^n}（1）$}保留，其中$ f^n：x^n \ to y^n $。在最后一节中，我们提出了一个问题，即“ hilali” - 类型不等式$ hp^π_x（r_x）\ leqq p_x（r_x）$是否具有合理的多重空间$ x $，前提
In this paper we introduce homological and homotopical Poincaré polynomials $P_f(t)$ and $P^π_f(t)$ of a continuous map $f:X \to Y$ such that if $f:X \to Y$ is a constant map, or more generally, if $Y$ is contractible, then these Poincaré polynomials are respectively equal to the usual homological and homotopical Poincaré polynomials $P_X(t)$ and $P^π_X(t)$ of the source space $X$. Our relative Hilali conjecture $P^π_f(1) \leqq P_f(1)$ is a map version of the the well-known Hilali conjecture $P^π_X(1) \leqq P_X(1)$ of a rationally elliptic space X. In this paper we show that under the condition that $H_i(f;\mathbb Q):H_i(X;\mathbb Q) \to H_i(Y;\mathbb Q)$ is not injective for some $i>0$, the relative Hilali conjecture of product of maps holds, namely, there exists a positive integer $n_0$ such that for $\forall n \geqq n_0$ the \emph{strict inequality $P^π_{f^n}(1) < P_{f^n}(1)$} holds, where $f^n:X^n \to Y^n$. In the final section we pose a question whether a "Hilali"-type inequality $HP^π_X(r_X) \leqq P_X(r_X)$ holds for a rationally hyperbolic space $X$, provided the the homotopical Hilbert--Poincare series $HP^π_X(r_X)$ converges at the radius $r_X$ of convergence.
