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Zhongjian Zhu

Publications and source records attributed to Zhongjian Zhu.

14 recordsLinked to original sources
arXiv2026

Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$

In this paper, we classify the homotopy types of the total spaces of $S^{2k-1}$-bundles (or fibrations) over $S^{2k}$ for $2\leq k\leq 6$. One of the two key new ingredients in the argument is the new necessary and sufficient conditions for a CW complex to be homotopy equivalent to the total space of a sphere bundle (fibration); the other is a formula relating the attaching map of the top cell of the total space and the characteristic map of a sphere bundle for $k=2,4$. When $k=4$, the classification results provide a negative answer to the conjecture in [6].

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arXiv2026

Complexes equivalent to $S^{2k-1}$-fibrations over $S^{2k}$

In this paper, necessary and sufficient conditions are obtained for the attaching map $f$ of the top cell of a CW complex to have the homotopy type of the total space of $S^{2k-1}$-fibration over $S^{2k}$ for any $k\geq 2$. As an application, the order of any attaching map of the top cell of the total space of an $S^{2k-1}$-fibration over $S^{2k}$ is determined and when $k\ne 2,4$, the homotopy types of the total spaces of $S^{2k-1}$-fibrations over $S^{2k}$ are classified by the stable homotopy classes of the attaching maps.

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arXiv2024

The unstable homotopy groups of 2-cell complexes

In this paper, we develop the new method, initiated by B. Gray (1972), to compute the unstable homotopy groups of the mapping cone, especially for $2$-cell complex $X=S^m\cup_α e^{n}$. By Gray's work mentioned above or the traditional method given by I.M.James (1957) which were widely used in previous related work to compute $π_{i}(X)$, the dimension $i\leq 2n+m-4$. By our method, we can compute $π_{i}(X)$ for $i>2n+m-4$. We use this different technique to generalize J.Wu's work, at Mem. of AMS, on homotopy groups of mod $2$ Moore spaces to higher dimensional homotopy groups of mod $2^r$ Moore spaces $P^{n}(2^r)$ for all $r\geq 1$. This practice shows that the technique given here is a new general method to compute the unstable homotopy groups of CW complexes with higher dimension.

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arXiv2024

The relative James construction and its application to homotopy groups

In this paper, we develop the new method to compute the homotopy groups of the mapping cone $C_f=Y\cup_{f}CX$ beyond the metastable range by analysing the homotopy of the $n$-th filtration of the relative James construction $J(X,A)$ for CW-pair $A\stackrel{i}\hookrightarrow X$, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map $X\cup_{i}CA\rightarrow ΣA$. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod $2^r$ Moore spaces for all positive integers $r$.

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arXiv2024

The homotopy decomposition of the suspension of a non-simply-connected $5$-manifold

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $5$-manifolds. As applications, we compute the reduced $K$-groups of $M$ and show that the suspension map between the third cohomotopy set $π^3(M)$ and the fourth cohomotopy set $π^4(ΣM)$ is a bijection.

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arXiv2024

The order of the group of self-homotopy equivalence of wedge spaces

In this paper $Aut(ΣX\vee ΣY)^\#$ the order of the group of self-homotopy equivalence of wedge spaces is studied. Under the condition of reducibility, we decompose $ Aut(\bigvee\limits_{t=1}^{k}X_{t})$ to the product of subgroups which generalizes the known results for $k=2$. Then we also give the formula for $ Aut(\bigvee\limits_{t=1}^{k}ΣX_{t})^{\#}$.

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arXiv2024

The $n+3$, $n+4$ dimensional homotopy groups of $\mathbf{A}_n^2$-complexes

In this paper, we calculate the $n+3$, $n+4$ dimensional homotopy groups of indecomposable $\mathbf{A}_n^2$-complexes after localization at 2. This job is seen as a sequel to P.J. Hilton's work on the $n+1,n+2$ dimensional homotopy groups of $\mathbf{A}_n^2$-complexes (1950-1951). The main technique used is analysing the homotopy property of $J(X,A)$, defined by B. Gray for a CW-pair $(X,A)$, which is homotopy equivalent to the homotopy fibre of the pinch map $X\cup CA\rightarrow ΣA$. By the way, the results of these homotopy groups have been used to make progress on recent popular topic about the homotopy decomposition of the (multiple) suspension of oriented closed manifolds.

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arXiv2024

The Homotopy Types of Suspended Simply-connected $6$-manifolds

Under the assumption that certain Adem cohomology operation acts trivially on $H^2(M;\mathbb{Z}/2)$, we determine the homotopy types of the triple suspension $Σ^3M$ of a simply-connected oriented closed topological(or smooth) $6$-manifold $M$, whose integral homology groups can have $2$-torsion.

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arXiv2023

2-local unstable homotopy groups of indecomposable $\mathbf{A}_3^2$ -complexes

In this paper, we calculate the 2-local unstable homotopy groups of indecomposable $\mathbf{A}_3^2$-complexes. The main technique used is analysing the homotopy property of $J(X,A)$, defined by B. Gray for a CW-pair $(X,A)$, which is homotopy equivalent to the homotopy fibre of the pinch map $X\cup CA\rightarrow ΣA$.

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arXiv2023

Suspension Homotopy of $(n-1)$-connected $(2n+2)$-dimensional Poincaré Duality Complexes

We study the homotopy decompositions of the suspension $ΣM$ of an $(n-1)$-connected $(2n+2)$ dimensional Poincaré duality complex $M$, $n\geq 2$. In particular, we completely determine the homotopy types of $ΣM$ of a simply-connected orientable closed (smooth) $6$-manifold $M$, whose integral homology groups can have $2$-torsion. If $3\leq n\leq 5$, we obtain homotopy decompositions of $ΣM$ after localization away from $2$.

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arXiv2017

Automorphisms of metacyclic groups

A metacyclic group $H$ can be presented as $\langle α,β\mid α^{n}=1, \ β^{m}=α^{t}, \ βαβ^{-1}=α^{r}\rangle$ for some $n,m,t,r$. Each endomorphism $σ$ of $H$ is determined by $σ(α)=α^{x_{1}}β^{y_{1}}, σ(β)=α^{x_{2}}β^{y_{2}}$ for some integers $x_{1},x_{2},y_{1},y_{2}$. We give sufficient and necessary conditions on $x_{1},x_{2},y_{1},y_{2}$ for $σ$ to be an automorphism.

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arXiv2015

The stable homotopy classification of $(n-1)$-connected $(n+4)$-dimensional polyhedra with 2 torsion free homology

In this paper, we study the stable homotopy types of $\mathbf{F}^4_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+4)$-dimensional polyhedra with 2-torsion free homologies. We are able to classify the indecomposable $\mathbf{F}^4_{n(2)}$-polyhedra. The proof relies on the matrix problem technique which was developed in the classification of representaions of algebras and applied to homotopy theory by Baues and Drozd.

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