An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

We prove the following two new optimal immersion results for complex projective space. First, if $n\equiv3\,\Mod 8$ but $n\not\equiv3\,\Mod 64$, and $\alpha(n)=7$, then $CP^{n}$ can be immersed in $\mathbb{R}^{4n-14}$. Second, if $n$ is even and $\alpha(n)=3$, then $CP^n$ can be immersed in $\mathbb{R}^{4n-4}$. Here $\alpha(n)$ denotes the number of 1s in the binary expansion of $n$. The first contradicts a result of Crabb, which said that such an immersion does not exist, apparently due to an arithmetical mistake. We combine Crabb's method with that developed by the author and Mahowald.

given a compact $n$-dimensional immersed riemannian manifold $m^n$ in some euclidean space we prove that if the hausdorff dimension of the singular set of the gauss map is small, then $m^n$ is homeomorphic to the sphere $s^n$.

also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with a small set of points of zero gauss–kronecker curvature are topologically the sphere minus a finite number of points. a characterization of the $2n$-catenoid is obtained.