Oriented bordism

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$\Omega_4^{SO}=\Zz$, generated by the complex projective space $\CP^2$.
$\Omega_4^{SO}=\Zz$, generated by the complex projective space $\CP^2$.
$\Omega_5^{SO}=\Zz_2$, generated by the Wu manifold.
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$\Omega_5^{SO}=\Zz_2$, generated by the Wu manifold $SU_3/SO_3$.
$\Omega_6^{SO}=\Omega_7^{SO}=0$.
$\Omega_6^{SO}=\Omega_7^{SO}=0$.
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== Invariants ==
== Invariants ==
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Signature. Pontryagin numbers. Stiefel-Whitney numbers.
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[[Wikipedia:Pontryagin class#Pontryagin numbers|Pontryagin numbers]], in particular the [[Wikipedia:Signature_(topology)|signature]].
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[[Wikipedia:Stiefel-Whitney class#Stiefel-Whitney numbers|Stiefel-Whitney numbers]].
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<wikitex>;
Thom {{cite|Thom1954}} computed $\Omega_*^{SO}\otimes \Qq$. This is equivalent to the computation of the rational (co)homology of $BSO$, as shown
Thom {{cite|Thom1954}} computed $\Omega_*^{SO}\otimes \Qq$. This is equivalent to the computation of the rational (co)homology of $BSO$, as shown
[[B-Bordism#Spectral sequences|here]]). The cohomology $H^*(BSO;\Qq)$ is a polynomial ring with generators the
+
[[B-Bordism#Spectral sequences|here]]. The cohomology $H^*(BSO;\Qq)$ is a polynomial ring with generators the
Pontryagin classes, so that Pontryagin numbers give an additive isomorphism $\Omega_*^{SO}\otimes \Qq \cong \Qq[x_{4i}]$.
Pontryagin classes, so that Pontryagin numbers give an additive isomorphism $\Omega_*^{SO}\otimes \Qq \cong \Qq[x_{4i}]$.
Since all products of $\CP^{2i}$ have linearly independent collections of Pontryagin numbers,
Since all products of $\CP^{2i}$ have linearly independent collections of Pontryagin numbers,
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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 $\Omega_*^{SO}$.
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== Further topics ==
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Revision as of 14:44, 3 February 2010

Contents

1 Introduction

By the Pontrjagin-Thom isomorphism the oriented bordism groups \Omega_n^{SO} of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum MSO.

2 Generators

\Omega_0^{SO}=\Zz, generated by a point.

\Omega_1^{SO}=0, as circles bound disks.

\Omega_2^{SO}=0, as oriented surfaces bound handlebodies.

\Omega_3^{SO}=0.

\Omega_4^{SO}=\Zz, generated by the complex projective space \CP^2.

\Omega_5^{SO}=\Zz_2, generated by the Wu manifold SU_3/SO_3.

\Omega_6^{SO}=\Omega_7^{SO}=0.

\Omega_*^{SO}\otimes \Qq is a polynomial ring, with generators \CP^{2i}.

3 Invariants

Pontryagin numbers, in particular the signature. Stiefel-Whitney numbers.

4 Classification

Thom [Thom1954] computed \Omega_*^{SO}\otimes \Qq. This is equivalent to the computation of the rational (co)homology of BSO, as shown here. The cohomology H^*(BSO;\Qq) is a polynomial ring with generators the Pontryagin classes, so that Pontryagin numbers give an additive isomorphism \Omega_*^{SO}\otimes \Qq \cong \Qq[x_{4i}]. Since all products of \CP^{2i} have linearly independent collections of Pontryagin numbers, there is a ring isomorphism from \Omega_*^{SO}\otimes \Qq to a polynomial ring with generators \CP^{2i}.

Averbuch, Milnor [Milnor1960], Thom showed that \Omega_*^{SO} has no odd torsion and \Omega_*^{SO}/\text{Torsion} is isomorphic to a polynomial ring \Zz[Y_{4i}]. Here the generators Y_{4i} can be any 4i-dimensional manifolds such that the Pontryagin number s_i(p_1,\dots p_k)(Y) equals \pm1, if 2k+1 is not a prime power, or equals \pm q, if 2k+1 is a power of the prime q. (Here s_i is the polynomial which expresses \sum t_k^i in terms of the elementary symmetric polynomials of the t_i.)

Wall [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 if and only if they have the same Stiefel-Whitney and Pontryagin numbers.

For the complete structure, we first describe the subalgebra \mathcal{W} of the unoriented bordism ring \mathcal{N} consisting of classes which contain a manifold M whose first Stiefel-Whitney class is the reduction of an integral class. \mathcal{W} is a polynomial ring on the following generators.

  • For k=2^{r-1}(2s+1) with integers r and s>0 (i.e. k not a power of 2), we have generators X_{2k-1}=P(2^r-1,2^rs), the Dold manifolds.
  • Reflection of S^{2^r-1} at the equator induces a map X_{2k-1}\to X_{2k-1}. The generator X_{2k} is the mapping torus of this map.
  • For k a power of 2, the generator X_k^2=\RP^k\times \RP^k.

Now there is an exact sequence

\displaystyle  \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} 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, this determines the ring structure of \Omega_*^{SO}.

5 References

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

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