Dold manifold

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