Dold manifold

In mathematics, a Dold manifold is one of the manifolds $P(m,n)=(S^m\times {\mathbb{ℂ}\mathbb{\mathbb{P}}}^n)/\tau$, where $\tau$ is the involution that acts as −1 on the m-sphere $S^m$ and as complex conjugation on the complex projective space ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$. These manifolds were constructed by Albrecht Dold (1956), who used them to give explicit generators for René Thom's unoriented cobordism ring. Note that $P(m,0)={\mathbb{ℝ}\mathbb{\mathbb{P}}}^m$, the real projective space of dimension m, and $P(0,n)={\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$.