# Dold manifold

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## 1 Introduction

A Dold manifold is a manifold of the form

$\displaystyle P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,$
where $m>0$$ {{Stub}} == Introduction == ; A Dold manifold is a manifold of the form P(m,n):= (S^m \times \mathbb {CP}^n)/\tau, where m>0, and the involution \tau sends (x,[y]) to (-x, [\bar y]) where \bar y = (\bar y_0,...,\bar y_n) for y =(y_0,...y_n). Dold used these manifolds in {{cite|Dold1956}} as generators for the [[Unoriented bordism|unoriented bordism ring]]. == Construction and examples == ; ... == Invariants == ; {{beginthm|Theorem {{cite|Dold1956}}}} The classes c \in H^1(P(m,n)) and d\in H^2(P(m,n)) generate H^*(P(m,n);\mathbb Z/2) with only the relations c^{m+1} =0 and d^{n+1} =0. The Steenrod squares act by Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2, and all other Squares Sq^i act trivially on c and d. On the decomposable classes the action is given by the Cartan formula. The total Stiefel-Whitney class of the tangent bundle is w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}. {{endthm}} == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]m>0$, and the involution $\tau$$\tau$ sends $(x,[y])$$(x,[y])$ to $(-x, [\bar y])$$(-x, [\bar y])$ where $\bar y = (\bar y_0,...,\bar y_n)$$\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$$y =(y_0,...y_n)$.

Dold used these manifolds in [Dold1956] as generators for the unoriented bordism ring.

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## 3 Invariants

Theorem [Dold1956] 3.1. The classes $c \in H^1(P(m,n))$$c \in H^1(P(m,n))$ and $d\in H^2(P(m,n))$$d\in H^2(P(m,n))$ generate $H^*(P(m,n);\mathbb Z/2)$$H^*(P(m,n);\mathbb Z/2)$ with only the relations

$\displaystyle c^{m+1} =0$

and

$\displaystyle d^{n+1} =0.$

The Steenrod squares act by

$\displaystyle Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,$

and all other Squares $Sq^i$$Sq^i$ act trivially on $c$$c$ and $d$$d$. On the decomposable classes the action is given by the Cartan formula.

The total Stiefel-Whitney class of the tangent bundle is

$\displaystyle w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}.$

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