Dold manifold

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1 Introduction

A Dold manifold is a manifold of the form

$\displaystyle P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,$ $$\displaystyle P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,$$

where $m>0$${{Stub}} == Introduction == <wikitex>; A Dold manifold is a manifold of the form $$ P(m,n):= (S^m \times \mathbb {CP}^n)/\tau, $$ where $m>0$, and the involution $\tau$ sends $(x,[y]) $ to $(-x, [\bar y])$ where $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$. Dold used these manifolds in {{cite|Dold1956}} as generators for the [[Unoriented bordism|unoriented bordism ring]]. </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; The fibre bundle $p:P(m,n) \to \mathbb {RP}^m$ has a section $s([x]) := [(x,[1,...,1])]$ and we consider the cohomology classes (always with $\mathbb Z/2$-coefficients) $$ 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}. $$ {{endthm}} </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> <!-- == Acknowledgments == ... == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]m>0$, and the involution $\tau$$\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.

2 Construction and examples

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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)

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

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

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

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

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

and

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

The Steenrod squares act by

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

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

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

4 Classification/Characterization

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5 Further discussion

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6 References

• [Dold1956] A. Dold, Erzeugende der Thomschen Algebra $\mathcal{N}$$\mathcal{N}$, Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601