Oriented bordism

Revision as of 16:53, 4 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|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 $\CP^2 \times \CP^2$. $\Omega_*^{SO}\otimes \Qq$ is a polynomial ring, with generators $\CP^{2i}$. </wikitex> == Invariants == <wikitex>; 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 n(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&Stasheff(1974)|Theorm 4.9, Lemma 17.3}}) and clearly additive. Hence obtain 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 a$M$ $$ \sigma(M) = \langle L(p_1(M), \dots , p_n(M)), [M] \rangle.$$ </wikitex> == Classification == <wikitex>; 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, Milnor {{cite|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 i$-dimensional manifolds such that the Pontryagin number $s_i(p_1,\dots p_k)(Y)$ equals $\pm1$, if k+1$ is not a prime power, or equals $\pm q$, if k+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 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 [[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 $$ \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to $$ 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 $\Omega_*^{SO}$. </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 plane $\CP^2$.

• $\Omega_5^{SO}=\Zz_2$, generated by the 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 $\CP^2 \times \CP^2$.

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

3 Invariants

The signature of a closed oriented manifold is a fundamental bordism invariant defining a ring homomorphism

$$\displaystyle \sigma : \Omega_*^{SO} \to \Zz.$$

(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 Pontryagin number $p_J$ of a closed, oriented manifold $M$ of dimension $4n(J)$ is the integer

$$\displaystyle \langle p_{j_1}(M) \cup p_{j_2}(M) \cup \dots \cup p_{j_n}(M), [M]\rangle \in \Zz$$

where $p_{k}$ is the k-the Pontrjagin of $M$ and $[M]$ its fundamental class. The 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 [Milnor&Stasheff(1974), Theorm 4.9, Lemma 17.3]) and clearly additive. Hence obtain homomorphisms

$$\displaystyle p_J : \Omega_{n(J)}^{SO} \to \Zz \text{~~and~~} w_J : \Omega_{n(J)}^{SO} \to \Zz/2.$$

By Hirzebruch's signature theorem [Hirzebruch1953], [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 a$M$

$$\displaystyle \sigma(M) = \langle L(p_1(M), \dots , p_n(M)), [M] \rangle.$$

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 the following theorems.

Theorem 4.2 [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:

$$\displaystyle [M_0] = [M_1] \in \Omega_n^{SO} ~~\Longleftrightarrow ~~ \ p_J(M_0) = p_J(M_1) ~~and~~ w_J(M_0) = w_J(M_1) ~~ \forall J.$$

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 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

$$\displaystyle \dots \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to \dots$$

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 $\Omega_*^{SO}$.

5 References

• [Hirzebruch1953] F. Hirzebruch, Über die quaternionalen projektiven Räume, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1953 (1953), 301–312 (1954). MR0065155 (16,389a) Zbl 0057.15503
• [Hirzebruch1995] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, Berlin, 1995. MR1335917 (96c:57002) Zbl 0843.14009
• [Milnor&Stasheff(1974)] Template:Milnor&Stasheff(1974)
• [Milnor1960] J. Milnor, On the cobordism ring $\Omega ^\ast$ 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