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Xiugui Liu

Publications and source records attributed to Xiugui Liu.

10 recordsLinked to original sources
arXiv2026

On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces

In this paper, we study the rational realization problem for the classifying space $\B(X)$. We prove that if $S^{2n}$ is realized as $\B(X)$ for a simply-connected space $X$, then $X$ is $\pi$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)$ for any simply-connected $\pi$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg--MacLane spaces $K(\Q^s,n)\times K(\Q^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)$ for any simply-connected $\pi$-finite space $X$. Moreover, we prove that if $r\geq 2$ and $n\geq 3$ and $K(\Q^r,n)$ is realized as $\B(X)$ for a $\pi$-finite space $X$, then $X\simeq_{\Q}K(\Q^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.

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arXiv2022

On the rational homotopical nilpotency index of principal bundles

Let $\rm{Aut}(p)$ denote the space of all self-fibre homotopy equivalences of a principal $G$-bundle $p: E\rightarrow X$ of simply connected CW complexes with $E$ finite. When $G$ is a compact connected topological group, we show that there exists an inequality $$n-{\rm N}(p)\leq {\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_0)\leq n$$ for any space $X$, where $n$ is the number of non-trivial rational homotopy groups of $G$ and ${\rm N}(p)$ is defined in Section 2. In particular, ${\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_{0})=n$ if $p$ is a fibre-homotopy trivial bundle and X is finite.

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arXiv2021

Formality on rationalizations of simply connected CW complexes

In this paper, we show that for a simply connected CW complex $Y$ with $H^{*}(Y;\mathbb{Q})$ of finite dimension, if $H^{*}(Y;\mathbb{Q})$ is concentrated in degrees $\leq 3$, then the rationalization $Y_\mathbb{Q}$ is formal. As an application, we show that the spatial realizations of simply connected Sullivan algebras with rational homology concentrated in degrees $\leq 3$ are formal.

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arXiv2020

Rational homotopy type of mapping spaces via cohomology algebras

In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra $H^*(X; \Q)$ and the rational homotopy type of $Y$. From this, we deduce the existence of H-structures on a component of the mapping space $\map(X, Y)$, assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we will show that $\map(X, Y; f)\simeq\map(X, Y; f')$ if the corresponding \emph{Maurer-Cartan elements} are connected by an algebra automorphism of $H^\ast(X, \Q)$.

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arXiv2010

On a spectral sequence for twisted cohomologies

Let ($Ω^{\ast}(M), d$) be the de Rham cochain complex for a smooth compact closed manifolds $M$ of dimension $n$. For an odd-degree closed form $H$, there are a twisted de Rham cochain complex $(Ω^{\ast}(M), d+H_\wedge)$ and its associated twisted de Rham cohomology $H^*(M,H)$. We show that there exists a spectral sequence $\{E^{p, q}_r, d_r\}$ derived from the filtration $F_p(Ω^{\ast}(M))=\bigoplus_{i\geq p}Ω^i(M)$ of $Ω^{\ast}(M)$, which converges to the twisted de Rham cohomology $H^*(M,H)$. We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.

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arXiv2009

Detection of some elements in the stable homotopy groups of spheres

In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres $π_{p^nq+2pq+q-3}S$ which is of order $p$ and is represented by $k_0h_{n} \in Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p)$ in the Adams spectral sequence, where $p\geq 5$ is an odd prime, $n\geq 3$ and $q=2(p-1)$. In the course of the proof, a new family of homotopy elements in $π_{\ast}V(1)$ which is represented by $β_{\ast}{i^{\prime}}_{\ast}i_{\ast}({h}_n)\in Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),\mathbb{Z}_p)$ in the Adams sequence is detected.

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arXiv2009

A product involving the $β$-family in stable homotopy theory

In the stable homotopy groups $π_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family $ξ_{m,n}$ for $n \geq m + 2 >5$. In this paper, the authors show that the composite $ξ_{m,n}β_{s}\in π_{q(p^n+p^m+sp+s)-5}(S)$ for $2 \leq s < p$ is non-trivial, where $q=2(p-1)$ and $β_s \in π_{q(sp+s-1)-2}(S)$ is the known $β$-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence.

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