Dold manifold

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Revision as of 15:52, 1 April 2011


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

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.

2 Construction and examples

...

3 Invariants

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}(1+d)^{n+1}.

4 Classification/Characterization

...

5 Further discussion

...

6 References

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