Dold manifold

In mathematics, a Dold manifold is one of the manifolds ${\displaystyle P(m,n)=(S^{m}\times \mathbb {CP} ^{n})/\tau }$, where ${\displaystyle \tau }$ is the involution that acts as −1 on the m-sphere ${\displaystyle S^{m}}$ and as complex conjugation on the complex projective space ${\displaystyle \mathbb {CP} ^{n}}$. These manifolds were constructed by Albrecht Dold (1956),^[1] who used them to give explicit generators for René Thom's unoriented cobordism ring.^[2] Note that ${\displaystyle P(m,0)=\mathbb {RP} ^{m}}$, the real projective space of dimension m, and ${\displaystyle P(0,n)=\mathbb {CP} ^{n}}$.^[3]