Oriented bordism

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This page has not been refereed. The information given here might be incomplete or provisional.


1 Introduction

By the 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.

2 Generators

  • \Omega_0^{SO}=\Zz, generated by a point.
  • \Omega_1^{SO}=0, as circles bound disks.
  • \Omega_3^{SO}=0.
  • \Omega_5^{SO}=\Zz_2, generated by the 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 [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``.

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&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), 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. 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 [Averbuh1959], Milnor [Milnor1960], Thom showed that \Omega_*^{SO} has no odd torsion and Novikov [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 4k-dimensional manifold 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 [Rokhlin1953], determined the structure of \Omega_*^{SO} completely. In particular he proved the following theorems.

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

Theorem 3.2 [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 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 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 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

\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

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