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{{Stub}}== Introduction ==
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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 $MSO$. 
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== Generators ==
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* $\Omega_0^{SO}=\Zz$, generated by a point. 

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

* $\Omega_2^{SO}=0$, as oriented [[Surfaces|surfaces]] bound [[Wikipedia:Handlebody#3-dimensional handlebodies|handlebodies]].

* $\Omega_3^{SO}=0$.

* $\Omega_4^{SO}=\Zz$, generated by the [[Wikipedia:Complex projective space|complex projective plane]] $\CP^2$.

* $\Omega_5^{SO}=\Zz_2$, generated by the [[1-connected 5-manifolds/1st Edition#Constructions and examples|Wu manifold]] $SU_3/SO_3$.

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

* $\Omega_8^{SO} \cong \Zz \oplus \Zz$ generated by $\CP^4$ and $\CP^2 \times \CP^2$.

$\Omega_*^{SO} \neq 0$ for $* \geq 9$: see also {{cite|Milnor&Stasheff1974|p. 203}}.

$\Omega_*^{SO}\otimes \Qq$ is a polynomial ring, with generators $\CP^{2i}$.
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== Invariants ==
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The [[Wikipedia:Signature_(topology)|signature]] of a closed oriented manifold is a fundamental bordism invariant defining a ring homomorphism
$$ \sigma : \Omega_*^{SO} \to \Zz.$$
(Note that manifolds of dimension not divisible by 4 have signature zero.)

For a muli-index $J = (j_1, \dots , j_n)$ of degree $n(J) : = \Sigma_i j_i$ the 
[[Wikipedia:Pontryagin class#Pontryagin numbers|Pontryagin number]] $p_J$ of a closed, oriented manifold $M$ of dimension $4n(J)$ is the integer
$$ \langle p_{j_1}(M) \cup p_{j_2}(M) \cup \dots \cup p_{j_n}(M), [M]\rangle \in \Zz $$
where $p_{k}$ is the k-the Pontrjagin of $M$ and $[M]$ its [[Wikipedia:Fundamental_class|fundamental class]]. The [[Wikipedia:Stiefel-Whitney class#Stiefel–Whitney numbers|Stiefel-Whitney numbers]] of $M$, $w_J(M) \in \Zz/2$, are defined similarly using Stiefel-Whitney classes.  These numbers are bordism invariants (see for example {{cite|Milnor&Stasheff1974|Theorm 4.9, Lemma 17.3}}) and clearly additive.  Hence we have homomorphisms
$$ p_J : \Omega_{n(J)}^{SO} \to \Zz \text{~~and~~} w_J : \Omega_{n(J)}^{SO} \to \Zz/2.$$
By Hirzebruch's [[Wikipedia:Genus_of_a_multiplicative_sequence#L_genus_and_the_Hirzebruch_signature_theorem|signature theorem]] {{cite|Hirzebruch1953}}, {{cite|Hirzebruch1995|Theorem 8.2.2}}, there is a certain rational polynomial in the Pontrjagin classes, called the L-polynomial $L(p_1, \dots, p_n)$, which computes the signature of M
$$ \sigma(M) = \langle L_n(p_1(M), \dots , p_n(M)), [M] \rangle.$$
For example: 
$$L_0 = 1, ~ L_1 = \frac{p_1}{3}, ~ L_2 = \frac{7p_2 - p_1^2}{45}, ~ L_3 = \frac{62p_3-13p_2p_1 + 2p_1^3}{3^3 \cdot 5 \cdot 7}.
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== Classification ==
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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
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 \cite{Averbuh1959}, Milnor {{cite|Milnor1960}}, Thom showed that $\Omega_*^{SO}$ has no odd torsion and Novikov \cite{Novikov1960} showed that $\Omega_*^{SO}/\text{Torsion}$ 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 {{cite|Wall1960}}, using earlier results of \cite{Rohlin1953}, determined the structure of $\Omega_*^{SO}$ completely.  In particular he proved the following theorems.

{{beginthm|Theorem|{{cite|Wall1960|Theorem 2}}}}
All torsion in $\Omega_*^{SO}$ is of exponent 2.
{{endthm}}

{{beginthm|Theorem|{{cite|Wall1960|Corollary 1}}}}
Two closed oriented n-manifolds $M_0$ and $M_1$ are oriented cobordant
if and only if they have the same Stiefel-Whitney and Pontryagin numbers:
$$ [M_0] = [M_1] \in \Omega_n^{SO} ~~\Longleftrightarrow ~~ \ p_J(M_0) = p_J(M_1) ~~and~~ w_J(M_0) = w_J(M_1) ~~ \forall J.$$
{{endthm}}

For the complete ring 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 [[Unoriented bordism#Construction and examples|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
$$ \dots \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to \dots $$
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 $\Omega_*^{SO}$.
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== References ==
{{#RefList:}}

[[Category:Manifolds]]
[[Category:Bordism]]