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Jianzhong Pan

Publications and source records attributed to Jianzhong Pan.

At least 19 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].

math.AT↗
arXiv2026

Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations

This paper investigates stable cohomotopy groups in codimensions two and three from complementary algebraic and geometric viewpoints. For general CW complexes, we give a complete characterization of stable cohomotopy in codimension two and a characterization in codimension three up to a $3$-primary parameter. Geometrically, we provide bordism-theoretic interpretations of these stable cohomotopy groups for oriented manifolds in codimension two and string manifolds in codimension three. As an application, we derive necessary and sufficient conditions for the existence of nowhere-vanishing sections of vector bundles, extending the foundational codimension-one results of Konstantis.

math.AT↗
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.

math.AT↗
arXiv2025

Cohomotopy Sets of $(n-1)$-connected $(2n+2)$-manifolds for small $n$

Let $M$ be a closed orientable $(n-1)$-connected $(2n+2)$-manifold, $n\geq 2$. In this paper we combine the Postnikov tower of spheres and the homotopy decomposition of the reduced suspension space $ΣM$ to investigate the cohomotopy sets $π^\ast(M)$ for $n=2,3,4$, under the assumption that $M$ has $2$-torsion-free homology. All cohomotopy sets $π^i(M)$ of such manifolds $M$ are characterized except $π^4(M)$ for $n=3,4$.

math.AT↗
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 $X\cup CA\rightarrow ΣA$.

math.AT↗
arXiv2022

The Stable Picard Group of A(n)

In this paper, we showed that the Stable Picard group of $A(n)$ for $n\geq 2$ is $\mathbb{Z}\oplus \mathbb{Z}$ by considering the endotrivial modules over $A(n)$. The proof relies on reductions from a Hopf algebra to its proper Hopf subalgebras.

math.AT↗
arXiv2022

On Modular Cohomotopy Groups

Let $p$ be a prime and let $π^n(X;\mathbb{Z}/p^r)=[X,M_n(\mathbb{Z}/p^r)]$ be the set of homotopy classes of based maps from CW-complexes $X$ into the mod $p^r$ Moore spaces $M_n(\mathbb{Z}/p^r)$ of degree $n$, where $\mathbb{Z}/p^r$ denotes the integers mod $p^r$. In this paper we firstly determine the modular cohomotopy groups $π^n(X;\mathbb{Z}/p^r)$ up to extensions by classical methods of primary cohomology operations and give conditions for the splitness of the extensions. Secondly we utilize some unstable homotopy theory of Moore spaces to study the modular cohomotopy groups; especially, the group $π^3(X;\mathbb{Z}_{(2)})$ with $\dim(X)\leq 6$ is determined.

math.AT↗
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.

math.AT↗
arXiv2009

Only rational homology spheres admit $Ω(f)$ to be union of DE attractors

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $Ω(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expanding maps are expanding on (co)homologies.

math.GT↗
arXiv2005

Gerbes and twisted orbifold quantum cohomology

In this article, we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.

math.AG↗
arXiv2004

The Generalized Liouville's Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, ω)$, their properties and a kind of classification of vector fields on the manifold, we generalize Liouville's theorem in classical mechanics to two sequences, the symplectic(-like) and the Hamiltonian-(like) Liouville's theorems. This also generalizes Noether's theorem, since the sequence of symplectic(-like) Liouville's theorems link to the cohomology directly.

math-ph↗
arXiv2004

Toroidal orbifolds, gerbes and group cohomology

We compute the integral cohomology of certain semi-direct products arising from a linear G-action on the n-torus, where G is a finite group. The main application is the complete calculation of torsion gerbes for certain six dimensional examples arising from string theory.

math.AT↗
arXiv2004

The Chen-Ruan Cohomology of Almost Contact Orbifolds

Comparing to the Chen-Ruan cohomology theory for the almost complex orbifolds, we study the orbifold cohomology theory for almost contact orbifolds. We define the Chen-Ruan cohomology group of any almost contact orbifold. Using the methods for almost complex orbifolds (see [2]), we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen-Ruan cup product for the Chen-Ruan cohomology. We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring. Finally we calculate two examples.

math.SG↗
arXiv2003

The Euler-Lagrange Cohomology Groups on Symplectic Manifolds

The definition and properties of the Euler-Lagrange cohomology groups $H^{2k-1}$, $1 \leqslant k \leqslant n$, on a symplectic manifold $({\cal M}^{2n},ω)$ are given and studied. For $k = 1$ and $k = n$, they are isomorphic to the corresponding de Rham cohomology groups $H_{dR}^1({\cal M}^{2n})$ and $H_{dR}^{2n-1}({\cal M}^{2n})$, respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomology groups on $({\cal M}^{2n},ω)$, in general. The general volume-preserving equations on $({\cal M}^{2n},ω)$ are also presented from cohomological point of view. In the special cases, these equations become the ordinary canonical equations in the Hamilton mechanics. Therefore, the Hamilton mechanics has been generalized via the cohomology.

physics.class-ph↗
arXiv2003

The Euler-Lagrange Cohomology and General Volume-Preserving Systems

We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold $(\mathcal{M}^{2n}, ω)$ and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important 2-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian $H$ are included as a special case with the 2-form $\frac{1}{n-1} H ω$. It is studied the other volume preserving systems on $({\cal M}^{2n}, ω)$. It is also explored the relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics.

hep-th↗