Dold manifold
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== Invariants == | == Invariants == | ||
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+ | 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 | {{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 | ||
$$ c^{m+1} =0 | $$ c^{m+1} =0 |
Revision as of 15:56, 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
Dold used these manifolds in [Dold1956] as generators for the unoriented bordism ring.
2 Construction and examples
...
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
4 Classification/Characterization
...
5 Further discussion
...
6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601