Dold manifold

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Contents

[edit] 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, 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 [Dold1956] as generators for the unoriented bordism ring.

[edit] 2 Construction and examples

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[edit] 3 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)

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

where x is a generator of 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.

Theorem [Dold1956] 3.1. 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

\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 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

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

[edit] 4 Classification/Characterization

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[edit] 5 Further discussion

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[edit] 6 References

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