Unoriented bordism
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− | {{Authors|Matthias Kreck}} | + | {{Authors|Matthias Kreck}}{{Stub}} |
== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | We denote the | + | We denote the unoriented bordism groups by $\mathcal N_i$. The sum of these groups |
$$ | $$ | ||
\mathcal N_* := \sum _i\mathcal N_i | \mathcal N_* := \sum _i\mathcal N_i | ||
$$ | $$ | ||
− | + | forms a ring under cartesian products of manifolds. Thom {{cite|Thom1954}} has shown that this ring is a polynomial ring over $\mathbb Z/2$ in variables $x_i$ for $i \ne 2^k -1$ and he has shown that for $i$ even one can take $\mathbb {RP}^i$ for $x_i$. Dold {{cite|Dold1956}} has constructed manifolds for $x_i$ with $i $ odd. | |
</wikitex> | </wikitex> | ||
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Dold constructs certain bundles over $\mathbb {RP}^m$ with fibre $\mathbb {CP}^n$ denoted by | Dold constructs certain bundles over $\mathbb {RP}^m$ with fibre $\mathbb {CP}^n$ denoted by | ||
$$ | $$ | ||
− | P(m,n):= (S^m \times \mathbb {CP}^ | + | P(m,n):= (S^m \times \mathbb {CP}^n)/\tau, |
− | $$ where $\tau$ is the involution mapping $(x,[y]) $ to $(-x, [\bar y])$ and $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$. These manifolds are now | + | $$ where $\tau$ is the involution mapping $(x,[y]) $ to $(-x, [\bar y])$ and $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$. These manifolds are now called Dold manifolds. |
Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $\mathcal N_*$. | Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $\mathcal N_*$. | ||
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− | {{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 | + | {{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 | ||
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and all other Squares $Sq^i$ act trivially on $c$ and $d$. On the decomposable classes the action is given by the Cartan formula. | 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 | ||
+ | $$ | ||
+ | w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}. | ||
+ | $$ | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
− | == Classification | + | == Classification == |
<wikitex>; | <wikitex>; | ||
− | + | To give explicit polynomial generators is useful information, if one wants to prove a formula like for example that $<w_n(M),[M]> = e(M)\,\, mod\,\, 2$, where $w_n$ is the $n$-th Stiefel-Whitney class of an $m$-dimensional manifold and $e(M)$ is the Euler characteristic, which one has to check on generators. But it does not help to classify manifolds up to bordism. There is an explicit answer to this question by Thom: | |
+ | {{beginthm|Theorem {{cite|Thom1954}}}} Two closed $n$-manifolds $M$ and $N$ are bordant if and only if all Stiefel-Whitney numbers agree: | ||
+ | $$ | ||
+ | <w_{i_1}\cup....\cup w_{i_k}(M), [M]> = <w_{i_1}\cup....\cup w_{i_k}(N), [N]> | ||
+ | $$ | ||
+ | for all partitions $i_1+...+i_k =n$. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
== Further discussion == | == Further discussion == | ||
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− | + | For $i= 2^r(2s+1)-1$ odd $i \ne 2^k-1$ the manifolds $P_i:= P(2^r-1,s2^r)$ are orientable and thus after choosing an orientation give an element in the oriented bordism group $\Omega_i$. Since $P_i$ admits an obvious orientation reversing diffeomorphism, these elements are $2$-torsion. Thus we obtain a subring in $\Omega_*$ isomorphic to $\mathbb Z/2[x_5, x_9, x_{11},...]$. For more information about $\Omega _*$ see the page on oriented bordism.</wikitex> | |
− | </wikitex> | + | |
== References == | == References == | ||
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[[Category:Manifolds]] | [[Category:Manifolds]] | ||
− | + | [[Category:Bordism]] |
Latest revision as of 17:55, 10 December 2010
The user responsible for this page is Matthias Kreck. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
We denote the unoriented bordism groups by . The sum of these groups
forms a ring under cartesian products of manifolds. Thom [Thom1954] has shown that this ring is a polynomial ring over in variables for and he has shown that for even one can take for . Dold [Dold1956] has constructed manifolds for with odd.
2 Construction and examples
Dold constructs certain bundles over with fibre denoted by
Using the results by Thom [Thom1954] Dold shows that these manifolds give ring generators of .
Theorem (Dold) [Dold1956] 2.1. For even set and for set . Then for
are polynomial generators of olver :
3 Invariants
To prove the Theorem Dold has to compute the characteristic numbers which according to Thom's theorem determine the bordism class. As a first step Dold computes the cohomology ring with -coeffcients. 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
To give explicit polynomial generators is useful information, if one wants to prove a formula like for example that , where is the -th Stiefel-Whitney class of an -dimensional manifold and is the Euler characteristic, which one has to check on generators. But it does not help to classify manifolds up to bordism. There is an explicit answer to this question by Thom:
Theorem [Thom1954] 4.1. Two closed -manifolds and are bordant if and only if all Stiefel-Whitney numbers agree:
for all partitions .
5 Further discussion
For odd the manifolds are orientable and thus after choosing an orientation give an element in the oriented bordism group . Since admits an obvious orientation reversing diffeomorphism, these elements are -torsion. Thus we obtain a subring in isomorphic to . For more information about see the page on oriented bordism.
6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502