Oriented bordism

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1 Introduction

By the Pontrjagin-Thom isomorphism the oriented bordism groups $\Omega_n^{SO}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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

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

(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

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

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

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

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

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

For example:

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

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:

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

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

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

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}$.

5 References

• [Averbuh1959] B. G. Averbuh, Algebraic structure of cobordism groups, Dokl. Akad. Nauk SSSR 125 (1959), 11–14. MR0124894 (23 #A2204)
• [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&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [Milnor1960] J. Milnor, On the cobordism ring $\Omega ^\ast$$\Omega ^\ast$ and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702
• [Novikov1960] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. MR0121815 (22 #12545) Zbl 0094.35902
• [Rokhlin1953] V. A. Rohlin, Intrinsic homologies, Doklady Akad. Nauk SSSR (N.S.) 89 (1953), 789–792. MR0056292 (15,53b)
• [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