Oriented bordism
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(Here $s_i$ is the polynomial which expresses $\sum t_k^i$ in terms of the elementary symmetric polynomials of the $t_i$.) | (Here $s_i$ is the polynomial which expresses $\sum t_k^i$ in terms of the elementary symmetric polynomials of the $t_i$.) | ||
− | Wall{{cite|Wall1960}}, using earlier results of Rohlin, determined the structure of $\Omega_*^{SO}$ completely. | + | Wall {{cite|Wall1960}}, using earlier results of Rohlin, determined the structure of $\Omega_*^{SO}$ completely. |
In particular he proved that all torsion in $\Omega_*^{SO}$ is of exponent 2, and that two manifolds are oriented cobordant | In particular he proved that all torsion in $\Omega_*^{SO}$ is of exponent 2, and that two manifolds are oriented cobordant | ||
if and only if they have the same Stiefel-Whitney and Pontryagin numbers. | if and only if they have the same Stiefel-Whitney and Pontryagin numbers. | ||
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$$ \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to $$ | $$ \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to $$ | ||
where the ring homomorphism $r$ is induced by the forgetful map $\Omega_q^{SO} \to \mathcal{N}$, and $r\partial:\mathcal{W}\to \mathcal{W}$ | where the ring homomorphism $r$ is induced by the forgetful map $\Omega_q^{SO} \to \mathcal{N}$, and $r\partial:\mathcal{W}\to \mathcal{W}$ | ||
− | is the derivation $X_{2k}\mapsto X_{2k-1}, X_{2k-1}\mapsto 0, X_k^2\mapsto 0$. | + | is the derivation $X_{2k}\mapsto X_{2k-1}, X_{2k-1}\mapsto 0, X_k^2\mapsto 0$. |
+ | |||
Together with the result that one can choose generators $Y_{4i}$ for $\Omega_*^{SO}/\text{Torsion}$ such that $r(Y_{4i})=X_{2i}^2$, | Together with the result that one can choose generators $Y_{4i}$ for $\Omega_*^{SO}/\text{Torsion}$ such that $r(Y_{4i})=X_{2i}^2$, | ||
this determines the ring structure of $\Omega_*^{SO}$. | this determines the ring structure of $\Omega_*^{SO}$. |
Revision as of 13:09, 3 February 2010
Contents |
1 Introduction
By the Pontrjagin-Thom isomorphism the oriented bordism groups of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum .
2 Generators
, generated by a point.
, as circles bound disks.
, as oriented surfaces bound handlebodies.
.
, generated by the complex projective space .
, generated by the Wu manifold.
.
is a polynomial ring, with generators .
3 Invariants
Signature. Pontryagin numbers. Stiefel-Whitney numbers.
4 Classification
Thom [Thom1954] computed . This is equivalent to the computation of the rational (co)homology of , as shown here). The cohomology is a polynomial ring with generators the Pontryagin classes, so that Pontryagin numbers give an additive isomorphism . Since all products of have linearly independent collections of Pontryagin numbers, there is a ring isomorphism from to a polynomial ring with generators .
Averbuch, Milnor [Milnor1960], Thom showed that has no odd torsion and is isomorphic to a polynomial ring . Here the generators can be any -dimensional manifolds such that the Pontryagin number equals , if is not a prime power, or equals , if is a power of the prime . (Here is the polynomial which expresses in terms of the elementary symmetric polynomials of the .)
Wall [Wall1960], using earlier results of Rohlin, determined the structure of completely. In particular he proved that all torsion in is of exponent 2, and that two manifolds are oriented cobordant if and only if they have the same Stiefel-Whitney and Pontryagin numbers.
For the complete structure, we first describe the subalgebra of the unoriented bordism ring consisting of classes which contain a manifold whose first Stiefel-Whitney class is the reduction of an integral class. is a polynomial ring on the following generators.
- For with integers and (i.e. not a power of 2), we have generators , the Dold manifolds.
- Reflection of at the equator induces a map . The generator is the mapping torus of this map.
- For a power of 2, the generator .
Now there is an exact sequence
where the ring homomorphism is induced by the forgetful map , and is the derivation .
Together with the result that one can choose generators for such that , this determines the ring structure of .
5 Further topics
6 References
- [Milnor1960] J. Milnor, On the cobordism ring and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
- [Wall1960] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR0120654 (22 #11403) Zbl 0097.38801
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