2023/08/08 by Bais, Valentina, Torres, Rafael · 2 citations
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2308.04227
Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere S inside ℝ P2× S2 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(2γ⊕ 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 ℝ P4. 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 ℝP4. 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.