Oriented bordism

Revision as of 21:24, 2 February 2010

1 Introduction

By the Pontrjagin-Thom isomorphism the oriented bordism groups $== Introduction == <wikitex>; By the [[B-Bordism#The Pontrjagin-Thom isomorphism|Pontrjagin-Thom isomorphism]] the oriented bordism groups $\Omega_n^{SO}$ of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum $MSO$. </wikitex> == Generators == <wikitex>; $\Omega_0^{SO}=\Zz$, 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 $\CP^2$. $\Omega_4^{SO}=\Zz_2$, generated by the Wu manifold. $\Omega_5^{SO}=\Omega_6^{SO}=\Omega_7^{SO}=0$. $\Omega_*^{SO}\otimes \Qq$ is a polynomial ring, with generators $\CP^{2i}$. </wikitex> == Invariants == <wikitex>; Signature. Pontryagin numbers. Stiefel-Whitney numbers. </wikitex> == Classification == <wikitex>; Thom {{cite|Thom1954}} computed $\Omega_*^{SO}\otimes \Qq$. This is equivalent to the computation of the rational (co)homology of $MSO$ (see [[B-Bordism#Spectral sequences|for the isomorphism]]). The cohomology $H^*(MSO;\Qq)$ is a polynomial ring with generators the Pontryagin classes, so that Pontryagin numbers give an additive isomorphism. Milnor, Rohlin, Wall{{cite|Wall1960}} </wikitex> == Further topics == <wikitex>; </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\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.

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

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

3 Invariants

Signature. Pontryagin numbers. Stiefel-Whitney numbers.

4 Classification

Thom [Thom1954] computed $\Omega_*^{SO}\otimes \Qq$. This is equivalent to the computation of the rational (co)homology of $MSO$ (see for the isomorphism). The cohomology $H^*(MSO;\Qq)$ is a polynomial ring with generators the Pontryagin classes, so that Pontryagin numbers give an additive isomorphism. Milnor, Rohlin, Wall[Wall1960]

5 Further topics

6 References

• [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