# Dold manifold

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

...

## 3 Invariants

The fibre bundle $p:P(m,n) \to \mathbb {RP}^m$$p:P(m,n) \to \mathbb {RP}^m$ has a section $s([x]) := [(x,[1,...,1])]$$s([x]) := [(x,[1,...,1])]$ and we consider the cohomology classes (always with $\mathbb Z/2$$\mathbb Z/2$-coefficients)

$\displaystyle c:= p^*(x) \in H^1(P(m,n)),$

where $x$$x$ is a generator of $H^1(\mathbb {RP}^m)$$H^1(\mathbb {RP}^m)$, and

$\displaystyle d \in H^2(P(m,n)),$

which is characterized by the property that the restriction to a fibre is non-trivial and $s^*(d)=0$$s^*(d)=0$.

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

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

...

...