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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.01979 |
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| _version_ | 1866911267408576512 |
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| author | Yang, Huijun |
| author_facet | Yang, Huijun |
| contents | Let $M$ be a closed oriented spin$^{c}$ manifold of dimension $(8n {+} 2)$ with fundamental class $[M]$, and let $ρ_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ denote the $\bmod ~ 2$ reduction homomorphism. For any torsion class $t \in H^{4n}(M;\mathbb{Z})$, we establish the identity \[ \langle ρ_2(t) \cdot Sq^2 ρ_2 (t), [M] \rangle = \langle ρ_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] 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 $β^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ 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 $β^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and consequently, $Sq^3 v_{4n}(M) = 0$, for any closed oriented spin$^{c}$ manifold $M$ with $\dim M \le 8n{+}1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Bilinear Form for Spin$^c$ Manifolds Yang, Huijun Algebraic Topology 57R20, 57R90 Let $M$ be a closed oriented spin$^{c}$ manifold of dimension $(8n {+} 2)$ with fundamental class $[M]$, and let $ρ_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ denote the $\bmod ~ 2$ reduction homomorphism. For any torsion class $t \in H^{4n}(M;\mathbb{Z})$, we establish the identity \[ \langle ρ_2(t) \cdot Sq^2 ρ_2 (t), [M] \rangle = \langle ρ_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] 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 $β^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ 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 $β^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and consequently, $Sq^3 v_{4n}(M) = 0$, for any closed oriented spin$^{c}$ manifold $M$ with $\dim M \le 8n{+}1$. |
| title | A Bilinear Form for Spin$^c$ Manifolds |
| topic | Algebraic Topology 57R20, 57R90 |
| url | https://arxiv.org/abs/2509.01979 |