arXiv subjects

Huijun Yang

Publications and source records attributed to Huijun Yang.

9 recordsLinked to original sources

arXiv2026

On the non-existence of almost complex structures on sphere bundles over complex projective spaces

We study the existence of almost complex structures on even-dimensional sphere bundles over complex projective spaces. For bundles $\xi_{n,q}$ with fibre $S^{2q}$ over $\mathbb{C} P^n$, we establish a necessary condition: if $q \ge a(n)$ for an explicit function, then the total space $E_{n,q}$ does not admit an almost complex structure. As an application, we analyse a concrete family associated with the canonical line bundle and obtain non-existence criteria in terms of $p$-adic valuations; for $p=2$ this yields a simple numerical bound. The proofs rely on Chern class computations and divisibility properties of characteristic classes. The results leave open the question of existence in the range $4 \le q < a(n)$.

math.AT↗

arXiv2025

A Bilinear Form for Spin$^c$ Manifolds

Let $M$ be a closed oriented spin$^{c}$ manifold of dimension $(8n {+} 2)$ with fundamental class $[M]$, and let $\rho_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ denote the $\bmod ~ 2$ reduction homomorphism. For any torsion class $t \in H^{4n}(M;\mathbb{Z})$, we establish the identity \[ \langle \rho_2(t) \cdot Sq^2 \rho_2 (t), [M] \rangle = \langle \rho_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] where $Sq^2$ is the Steenrod square, $v_{4n}(M)$ is the $4n$-th Wu class of $M$, $ x\cdot y$ denotes the cup product of $x$ and $y$, and $\langle \cdot ~, ~\cdot \rangle$ denotes the Kronecker product. This result generalizes the work of Landweber and Stong from spin to spin$^c$ manifolds. As an application, let $\beta^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ be the Bockstein homomorphism associated to the short exact sequence of coefficients $\mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2$. We deduce that $\beta^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and consequently, $Sq^3 v_{4n}(M) = 0$, for any closed oriented spin$^{c}$ manifold $M$ with $\dim M \le 8n{+}1$.

math.AT↗

arXiv2022

Turning vector bundles

We define a turning of a rank-$2k$ vector bundle $E \to B$ to be a homotopy of bundle automorphisms $\psi_t$ from $\mathbb{Id}_E$, the identity of $E$, to $-\mathbb{Id}_E$, minus the identity, and call a pair $(E, \psi_t)$ a turned bundle. We investigate when vector bundles admit turnings and develop the theory of turnings and their obstructions. In particular, we determine which rank-$2k$ bundles over the $2k$-sphere are turnable. If a bundle is turnable, then it is orientable. In the other direction, complex bundles are turned bundles and for bundles over finite $CW$-complexes with rank in the stable range, Bott's proof of his periodicity theorem shows that a turning of $E$ defines a homotopy class of complex structure on $E$. On the other hand, we give examples of rank-$2k$ bundles over $2k$-dimensional spaces, including the tangent bundles of some $2k$-manifolds, which are turnable but do not admit a complex structure. Hence turned bundles can be viewed as a generalisation of complex bundles. We also generalise the definition of turning to other settings, including other paths of automorphisms, and we relate the generalised turnability of vector bundles to the topology of their gauge groups and the computation of certain Samelson products.

math.GT↗

arXiv2020

The existence of contact structures on 9-manifolds

We give necessary and sufficient conditions for a closed orientable 9-manifold M to admit an almost contact structure. The conditions are stated in terms of the Stiefel-Whitney classes of M and other more subtle homotopy invariants of M. By a fundamental result of Borman, Eliashberg and Murphy, M admits an almost contact structure if and only if M admits an over-twisted contact structure. Hence we give necessary and sufficient conditions for M to admit an over-twisted contact structure and we prove that if N is another closed 9-manifold which is homotopy equivalent to M, then M admits an over-twisted contact structure if and only if N does. In addition, for W_i(M) the i-th integral Stiefel-Whitney class of M, we prove that if W_3(M) = 0 then W_7(M) = 0.

math.SG↗

arXiv2020

Topological classification of complex vector bundles over $8$-dimensional spin$^{c}$ manifolds

In this paper, complex vector bundles of rank $r$ over $8$-dimensional spin$^{c}$ manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an application, we got that two rank $3$ complex vector bundles over $4$-dimensional complex projective spaces $\C P^{4}$ are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank $3$ complex vector bundles over $\C P^{4}$ are determined. Combing Thomas's and Switzer's results with our work, we can assert that complex vector bundles over $\C P^{4}$ are all classified.

math.AT↗

arXiv2019

Stable almost complex structures on certain $10$-manifolds

Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $$\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in ρ_{2} ( TH^{4}(M;\Z) ) \}.$$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset ρ_{2}( H^{2}(M; \Z) )$. Then the necessary and sufficient conditions for $M$ to admit a stable almost complex structure are determined in terms of the characteristic classes and cohomology ring of $M$.

math.DG↗

arXiv2018

Connected sums of almost complex manifolds

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le α$, we prove that $\left(\sharp_{i=1}^α M_{i}\right) \sharp (α{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $α$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we get that $M\sharp \overline{\mathbb{C} P^{n}}$ also admits an almost complex structure. At last, as an application, we obtain that $α\mathbb{C} P^{2n}\sharp β\overline{\mathbb{C} P^{2n}}$ admits an almost complex structure if and only if $α$ is odd.

math.DG↗

arXiv2016

Some remarks on stable almost complex structures on manifolds

Let $X$ be an $(8k+i)$-dimensional pathwise connected $CW$-complex with $i=1$ or $2$ and $k\ge0$, $ξ$ be a real vector bundle over $X$. Suppose that $ξ$ admits a stable complex structure over the $8k$-skeleton of $X$. Then we get that $ξ$ admits a stable complex structure over $X$ if the Steenrod square $$\mathrm{Sq}^{2}\colon H^{8k-1}(X;\mathbb{Z}/2)\rightarrow H^{8k+1}(X;\mathbb{Z}/2)$$ is surjective. As an application, let $M$ be a $10$-dimensional manifold with no $2$-torsion in $H_{i}(M;\mathbb{Z})$ for $i=1,2,3$, and no $3$-torsion in $H_{1}(M;\mathbb{Z})$. Suppose that the Steenrod square $$\mathrm{Sq}^{2}\colon H^{7}(M;\mathbb{Z}/2)\rightarrow H^{9}(M;\mathbb{Z}/2)$$ is surjective. Then the necessary and sufficient conditions for the existence of a stable almost complex structure on $M$ are given in terms of the cohomology ring and characteristic classes of $M$.

math.AT↗

arXiv2011

Almost Complex Structures on (n-1)-connected 2n-manifolds

Let M be a closed (n-1)-connected 2n-dimensional smooth manifold with n > 2. In terms of the system of invariants for such manifolds introduced by Wall, we obtain necessary and sufficient conditions for M to admit an almost complex structure.

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