manifoldsodddimensional

An odd-dimensional manifold, or an odd-dimensional manifold, is a topological or differentiable manifold whose dimension n is odd, meaning n = 2k + 1 for some integer k ≥ 0. Locally, these spaces resemble R^{2k+1}, and the chosen level of structure (topological, smooth, analytic, or PL) determines what kinds of transition maps are allowed. They can be compact or noncompact and oriented or nonoriented, depending on the category considered.

Prominent examples include the odd spheres S^{2k+1}, real projective spaces RP^{2k+1}, and common Lie groups such as SO(3) and SU(2) (the latter being diffeomorphic to S^3). Odd-dimensional manifolds also arise naturally in geometry and physics. Geometry in odd dimensions often centers on contact structures, which require odd dimension. A standard contact form exists on R^{2n+1} and on S^{2n+1}, where the kernel of the form defines a maximally nonintegrable hyperplane field.

Topological consequences of odd dimension are notable. If a compact oriented odd-dimensional manifold M is considered, its Euler characteristic χ(M) is zero. By the relation between the Euler class and χ, this implies the tangent bundle of M admits a nowhere-vanishing vector field, so M has at least one nonzero tangent direction everywhere. Poincaré duality also imposes symmetries on the Betti numbers, reflecting the odd dimension.

In summary, odd-dimensional manifolds form a natural setting for geometric structures such as contact geometry, admit distinctive topological constraints like vanishing Euler characteristic in the compact oriented case, and include many familiar spaces arising in geometry and mathematical physics.