# Oriented bordism

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## 1 Introduction

By the Pontrjagin-Thom isomorphism the oriented bordism groups $\Omega_n^{SO}$${{Stub}}== Introduction == ; 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. == Generators == ; * \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, detected by the signature. * \Omega_5^{SO}=\Zz_2, generated by the [[1-connected 5-manifolds/1st Edition#Constructions and examples|Wu manifold]] SU_3/SO_3, detected by the deRham invariant. * \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}, detected by the Pontrjagin numbers. \Omega_*^{SO}/\text{Tors} is an integral polynomial ring with generators the Milnor hypersurfaces. == Invariants == ; 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&Stasheff1974|Theorm 4.9, Lemma 17.3}}) and clearly additive. Hence we have homomorphisms p_J : \Omega_{n(J)}^{SO} \to \Zz \quad \text{and} \quad 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}, L_4 = \frac{381p_4 - 71p_3p_1 - 19p_2^2 + 22p_2p_1^2 - 3p^4}{3^4 \cdot 5^2 \cdot 7}. == Classification == ; 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}. Independently Averbuch \cite{Averbuh1959} and Milnor {{cite|Milnor1960}} showed that \Omega_*^{SO} has no odd torsion. In addition, Novikov \cite{Novikov1960} showed that \Omega_*^{SO}/\text{Torsion} is isomorphic to a polynomial ring \Zz[Y_4,Y_8,Y_{12}, \dots ]. Here a generator Y_{4k} can be any k-dimensional manifold 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 \cite{Rokhlin1953}, 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 with coefficients \Zz/2 on generators X_j where neither j nor j+1 are powers of 2, together with generators X_k^2 where k is a power of 2. These generators can be described explicitly as follows: * For k=2^{r-1}(2s+1) with integers r and s>0 (i.e. k not a power of 2), the generator X_{2k-1} is the [[Unoriented bordism#Construction and examples|Dold manifold]] P(2^r-1,2^rs) and the generator X_{2k} is the mapping torus of the map X_{2k-1}\to X_{2k-1} given by the reflection of S^{2^r-1} at the equator. *For k not a power of 2, the generator X_{2k} is the mapping torus of a certain involution A: X_{2k-1}\to X_{2k-1}. Indeed any [[Unoriented bordism#Construction and examples|Dold manifold]] P(m,n) = (S^m \times \C P^n)/\tau has the involution A[(x_0, \ldots, x_{m-1},x_m),z] = A[(x_0, \ldots, x_{m-1},-x_m),z]. * For k a power of 2, the generator X_k^2 is \RP^k\times \RP^k. This generator is also represented by \CP^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}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Bordism]]\Omega_n^{SO}$ of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum $MSO$$MSO$.

## 2 Generators

• $\Omega_0^{SO}=\Zz$$\Omega_0^{SO}=\Zz$, generated by a point.
• $\Omega_1^{SO}=0$$\Omega_1^{SO}=0$, as circles bound disks.
• $\Omega_2^{SO}=0$$\Omega_2^{SO}=0$, as oriented surfaces bound handlebodies.
• $\Omega_3^{SO}=0$$\Omega_3^{SO}=0$.
• $\Omega_4^{SO}=\Zz$$\Omega_4^{SO}=\Zz$, generated by the complex projective plane $\CP^2$$\CP^2$, detected by the signature.
• $\Omega_5^{SO}=\Zz_2$$\Omega_5^{SO}=\Zz_2$, generated by the Wu manifold $SU_3/SO_3$$SU_3/SO_3$, detected by the deRham invariant.
• $\Omega_6^{SO}=\Omega_7^{SO}=0$$\Omega_6^{SO}=\Omega_7^{SO}=0$.
• $\Omega_8^{SO} \cong \Zz \oplus \Zz$$\Omega_8^{SO} \cong \Zz \oplus \Zz$ generated by $\CP^4$$\CP^4$ and $\CP^2 \times \CP^2$$\CP^2 \times \CP^2$. $\Omega_*^{SO} \neq 0$$\Omega_*^{SO} \neq 0$ for $* \geq 9$$* \geq 9$: see also [Milnor&Stasheff1974, p. 203]. $\Omega_*^{SO}\otimes \Qq$$\Omega_*^{SO}\otimes \Qq$ is a polynomial ring, with generators $\CP^{2i}$$\CP^{2i}$, detected by the Pontrjagin numbers. $\Omega_*^{SO}/\text{Tors}$$\Omega_*^{SO}/\text{Tors}$ is an integral polynomial ring with generators the Milnor hypersurfaces.

## 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)$$J = (j_1, \dots , j_n)$ of degree $n(J) : = \Sigma_i j_i$$n(J) : = \Sigma_i j_i$ the Pontryagin number $p_J$$p_J$ of a closed, oriented manifold $M$$M$ of dimension $4n(J)$$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}$$p_{k}$ is the k-the Pontrjagin of $M$$M$ and $[M]$$[M]$ its fundamental class. The Stiefel-Whitney numbers of $M$$M$, $w_J(M) \in \Zz/2$$w_J(M) \in \Zz/2$, are defined similarly using Stiefel-Whitney classes. These numbers are bordism invariants (see for example [Milnor&Stasheff1974, Theorm 4.9, Lemma 17.3]) and clearly additive. Hence we have homomorphisms $\displaystyle p_J : \Omega_{n(J)}^{SO} \to \Zz \quad \text{and} \quad w_J : \Omega_{n(J)}^{SO} \to \Zz/2.$

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)$$L(p_1, \dots, p_n)$, which computes the signature of M $\displaystyle \sigma(M) = \langle L_n(p_1(M), \dots , p_n(M)), [M] \rangle.$

For example: $\displaystyle L_0 = 1, ~ L_1 = \frac{p_1}{3}, ~ L_2 = \frac{7p_2 - p_1^2}{45}, ~ L_3 = \frac{62p_3-13p_2p_1 + 2p_1^3}{3^3 \cdot 5 \cdot 7}, L_4 = \frac{381p_4 - 71p_3p_1 - 19p_2^2 + 22p_2p_1^2 - 3p^4}{3^4 \cdot 5^2 \cdot 7}.$

## 4 Classification

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

Independently Averbuch [Averbuh1959] and Milnor [Milnor1960] showed that $\Omega_*^{SO}$$\Omega_*^{SO}$ has no odd torsion. In addition, Novikov [Novikov1960] showed that $\Omega_*^{SO}/\text{Torsion}$$\Omega_*^{SO}/\text{Torsion}$ is isomorphic to a polynomial ring $\Zz[Y_4,Y_8,Y_{12}, \dots ]$$\Zz[Y_4,Y_8,Y_{12}, \dots ]$. Here a generator $Y_{4k}$$Y_{4k}$ can be any $4k$$4k$-dimensional manifold such that the Pontryagin number $s_i(p_1,\dots, p_k)(Y)$$s_i(p_1,\dots, p_k)(Y)$ equals $\pm1$$\pm1$, if $2k+1$$2k+1$ is not a prime power, or equals $\pm q$$\pm q$, if $2k+1$$2k+1$ is a power of the prime $q$$q$. (Here $s_i$$s_i$ is the polynomial which expresses $\sum t_k^i$$\sum t_k^i$ in terms of the elementary symmetric polynomials of the $t_i$$t_i$.)

Wall [Wall1960], using earlier results of [Rokhlin1953], determined the structure of $\Omega_*^{SO}$$\Omega_*^{SO}$ completely. In particular he proved the following theorems.

Theorem 3.1 [Wall1960, Theorem 2]. All torsion in $\Omega_*^{SO}$$\Omega_*^{SO}$ is of exponent 2.

Theorem 3.2 [Wall1960, Corollary 1]. Two closed oriented n-manifolds $M_0$$M_0$ and $M_1$$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}$$\mathcal{W}$ of the unoriented bordism ring $\mathcal{N}$$\mathcal{N}$ consisting of classes which contain a manifold $M$$M$ whose first Stiefel-Whitney class is the reduction of an integral class. $\mathcal{W}$$\mathcal{W}$ is a polynomial ring with coefficients $\Zz/2$$\Zz/2$ on generators $X_j$$X_j$ where neither $j$$j$ nor $j+1$$j+1$ are powers of 2, together with generators $X_k^2$$X_k^2$ where $k$$k$ is a power of 2. These generators can be described explicitly as follows:

• For $k=2^{r-1}(2s+1)$$k=2^{r-1}(2s+1)$ with integers $r$$r$ and $s>0$$s>0$ (i.e. $k$$k$ not a power of 2), the generator $X_{2k-1}$$X_{2k-1}$ is the Dold manifold $P(2^r-1,2^rs)$$P(2^r-1,2^rs)$ and the generator $X_{2k}$$X_{2k}$ is the mapping torus of the map $X_{2k-1}\to X_{2k-1}$$X_{2k-1}\to X_{2k-1}$ given by the reflection of $S^{2^r-1}$$S^{2^r-1}$ at the equator.
• For $k$$k$ not a power of 2, the generator $X_{2k}$$X_{2k}$ is the mapping torus of a certain involution $A: X_{2k-1}\to X_{2k-1}$$A: X_{2k-1}\to X_{2k-1}$. Indeed any Dold manifold $P(m,n) = (S^m \times \C P^n)/\tau$$P(m,n) = (S^m \times \C P^n)/\tau$ has the involution $A[(x_0, \ldots, x_{m-1},x_m),z] = A[(x_0, \ldots, x_{m-1},-x_m),z]$$A[(x_0, \ldots, x_{m-1},x_m),z] = A[(x_0, \ldots, x_{m-1},-x_m),z]$.
• For $k$$k$ a power of 2, the generator $X_k^2$$X_k^2$ is $\RP^k\times \RP^k$$\RP^k\times \RP^k$. This generator is also represented by $\CP^k$$\CP^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$$r$ is induced by the forgetful map $\Omega_q^{SO} \to \mathcal{N}$$\Omega_q^{SO} \to \mathcal{N}$, and $r\partial:\mathcal{W}\to \mathcal{W}$$r\partial:\mathcal{W}\to \mathcal{W}$ is the derivation $X_{2k}\mapsto X_{2k-1}, X_{2k-1}\mapsto 0, X_k^2\mapsto 0$$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}$$Y_{4i}$ for $\Omega_*^{SO}/\text{Torsion}$$\Omega_*^{SO}/\text{Torsion}$ such that $r(Y_{4i})=X_{2i}^2$$r(Y_{4i})=X_{2i}^2$, this determines the ring structure of $\Omega_*^{SO}$$\Omega_*^{SO}$.