Rigidity of self-maps of $V_{n,2}$ and classification of manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914483101761536 |
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| author | Biswas, Sagnik |
| author_facet | Biswas, Sagnik |
| contents | We study two problems concerning the Stiefel manifolds $V_{n,2}$ and their products with spheres. First, we address a rigidity problem: we determine, for most values of~$n$, all self-maps of $V_{n,2}$ that are homotopic to an almost diffeomorphism. Second, we classify smooth closed manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$ up to almost diffeomorphism, for $k = 3, 5$ or $7 \leq k \leq n-3$, $k \neq 2^i - 2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences. In favourable cases -- notably $V_{12,2} \times S^3$, $V_{16,2} \times S^3$, $V_{12,2} \times S^5$, $V_{10,2} \times S^5$ -- the classification is complete: every such manifold is almost diffeomorphic to $V_{n,2} \mathbin{\#} Σ\times S^k$ for some exotic sphere $Σ$. In the general case, we identify inverses for a large subgroup of $\operatorname{Im}(η)$ and provide a possible way forward to the remainder. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity of self-maps of $V_{n,2}$ and classification of manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$ Biswas, Sagnik Algebraic Topology Geometric Topology 57-xx(Primary), 55-xx(Secondary) We study two problems concerning the Stiefel manifolds $V_{n,2}$ and their products with spheres. First, we address a rigidity problem: we determine, for most values of~$n$, all self-maps of $V_{n,2}$ that are homotopic to an almost diffeomorphism. Second, we classify smooth closed manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$ up to almost diffeomorphism, for $k = 3, 5$ or $7 \leq k \leq n-3$, $k \neq 2^i - 2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences. In favourable cases -- notably $V_{12,2} \times S^3$, $V_{16,2} \times S^3$, $V_{12,2} \times S^5$, $V_{10,2} \times S^5$ -- the classification is complete: every such manifold is almost diffeomorphic to $V_{n,2} \mathbin{\#} Σ\times S^k$ for some exotic sphere $Σ$. In the general case, we identify inverses for a large subgroup of $\operatorname{Im}(η)$ and provide a possible way forward to the remainder. |
| title | Rigidity of self-maps of $V_{n,2}$ and classification of manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$ |
| topic | Algebraic Topology Geometric Topology 57-xx(Primary), 55-xx(Secondary) |
| url | https://arxiv.org/abs/2604.15984 |