Dold manifold
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Contents |
[edit] 1 Introduction
A Dold manifold is a manifold of the form
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 has a section and we consider the cohomology classes (always with -coefficients)
where is a generator of , and
which is characterized by the property that the restriction to a fibre is non-trivial and .
Theorem [Dold1956] 3.1. The classes and generate with only the relations
and
The Steenrod squares act by
and all other Squares act trivially on and . On the decomposable classes the action is given by the Cartan formula.
The total Stiefel-Whitney class of the tangent bundle is
[edit] 4 Classification/Characterization
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[edit] 5 Further discussion
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[edit] 6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601