تقرير
Smooth structures on non-orientable $4$-manifolds via twisting operations
العنوان: | Smooth structures on non-orientable $4$-manifolds via twisting operations |
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المؤلفون: | Bais, Valentina, Torres, Rafael |
سنة النشر: | 2023 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Geometric Topology |
الوصف: | Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is homeomorphic but not diffeomorphic to the total space of the non-trivial 2-sphere bundle over the real projective plane $S(2\gamma \oplus \mathbb{R})$. The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold $Y$ and a mapping torus that was used by Cappell-Shaneson to construct an exotic $\mathbb{R} P^4$. This construction of $Y$ is similar to the one of the Cappell-Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside $Y$ produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of $\mathbb{R}P^4$. Knotting phenomena of 2-spheres in non-orientable 4-manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fourth observation. |
نوع الوثيقة: | Working Paper |
URL الوصول: | http://arxiv.org/abs/2308.04227 |
رقم الأكسشن: | edsarx.2308.04227 |
قاعدة البيانات: | arXiv |
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