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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 $\Sigma M$ to investigate the cohomotopy sets $\pi^\ast(M)$ for $n=2,3,4$, under the assumption that $M$ has $2$-torsion-free homology. All cohomotopy sets $\pi^i(M)$ of such manifolds $M$ are characterized except $\pi^4(M)$ for $n=3,4$., Comment: 31 pages

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 $X\cup CA\rightarrow \Sigma A$.

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., Comment: 20 pages

Let $p$ be a prime and let $\pi^n(X;\mathbb{Z}/p^r)=[X,M_n(\mathbb{Z}/p^r)]$ 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 $\pi^n(X;\mathbb{Z}/p^r)$ 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 $\pi^3(X;\mathbb{Z}_{(2)})$ with $\dim(X)\leq 6$ is determined., Comment: 23 pages

In this paper, we classify the congruence classes of $\mathbf{F}^5_{n(2)}$-polyhedra, i.e. $(n-1)$-connected, at most $(n+5)$-dimensional polyhedra with 2-torsion free homology. The proof relies on the matrix problem technique which was developed in the classification of representations of algebras and applied to homotopy theory by Baues and Drozd., Comment: This article is accepted by SCIENCE CHINA Mathematics. arXiv admin note: text overlap with arXiv:1509.07932

In this paper, we determine the decomposability of smash product of two indecomposable A_n^2-complexes, i.e., (n-1)-connected finite CW-complexes with dimension at most n+2 (n\geq 3).

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.

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $\Omega(f)$ 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., Comment: 23 pages, 2 figures

We compute the cohomology of crystallographic groups with holonomy of prime order. As an application we compute the group of gerbes associated to many six--dimensional toroidal orbifolds arising in string theory.

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., Comment: Amstex

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, \omega)$, 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., Comment: 27 pages, no figure, revtex4

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.

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., Comment: 11 pages

The definition and properties of the Euler-Lagrange cohomology groups $H^{2k-1}$, $1 \leqslant k \leqslant n$, on a symplectic manifold $({\cal M}^{2n},\omega)$ 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 $({\cal M}^{2n},\omega)$, in general. The general volume-preserving equations on $({\cal M}^{2n},\omega)$ 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., Comment: 20 pages, no figures

We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold $(\mathcal{M}^{2n}, \omega)$ 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 $\frac{1}{n-1} H \omega$. It is studied the other volume preserving systems on $({\cal M}^{2n}, \omega)$. It is also explored the relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics., Comment: Plain LaTeX, use packages amssymb and amscd, 15 pages, no figures

In our previous work, a relation between Tsukiyama problem about self homotopy equivalence was found by using a generalization of phantom map. In this note, fundamental result is established for such a generalization. This is the first time one can deal with phantom maps to space not satisfying finite type condition. Application to Forgetful map is also discussed briefly., Comment: 10 pages

In this paper, we prove that the Mislin genus of a (co-)H-map between (co-)H-spaces under certain natural conditions is a finite abelian group which generalizes results in Zabrodsky, McGibbon and Hurvitz, Comment: 27 pages

In this note, we answer in negative a question posed by McGibbon about the generic property of H-space structure. In fact we verify the conjecture of Roitberg. Incidentally, the same example also answers in negative the open problem 10 in McGibbon.

A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou.

In this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact nonnegative curved manifolds admit (complete) metrics with nonnegative curvature., Comment: 7 pages, talk to be given at SISTAG, National Univ. of Singapore, Jul.2-6, 2001

In this paper we define a secondary Brown-Kervaire quadratic forms. Among the applications we obtain a complete classification of (n-2)-connected 2n-dimensional framed manifolds up to homeomorphism and homotopy equivalence, . In particular, we prove that the homotopy type of such manifolds determine their homeomorphism type.