Oriented bordism
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1 Introduction
By the Pontrjagin-Thom isomorphism the oriented bordism groups of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum
.
2 Generators
-
, generated by a point.
-
, as circles bound disks.
-
, as oriented surfaces bound handlebodies.
-
.
-
, generated by the complex projective plane
, detected by the signature.
-
, generated by the Wu manifold
, detected by the deRham invariant.
-
.
-
generated by
and
.
for
: see also [Milnor&Stasheff1974, p. 203].
is a polynomial ring, with generators
, detected by the Pontrjagin numbers.
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.](/images/math/f/5/5/f55abff948d835a30c4bb87c293d89fe.png)
(Note that manifolds of dimension not divisible by 4 have signature zero.)
For a muli-index of degree
the
Pontryagin number
of a closed, oriented manifold
of dimension
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](/images/math/d/5/a/d5a0ef40d8e53b9b8a4e421d1ea01adb.png)
where is the k-the Pontrjagin of
and
its fundamental class. The Stiefel-Whitney numbers of
,
, 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 \text{~~and~~} w_J : \Omega_{n(J)}^{SO} \to \Zz/2.](/images/math/1/9/3/193b196bf784fb6697ee792ed97d5a3f.png)
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 , which computes the signature of M
![\displaystyle \sigma(M) = \langle L_n(p_1(M), \dots , p_n(M)), [M] \rangle.](/images/math/2/4/f/24f787ebe0172284dfde21df35d8cc3d.png)
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}.](/images/math/d/9/6/d96367b41be6a4704bb36f397e291ad6.png)
4 Classification
Thom [Thom1954] computed . This is equivalent to the computation of the rational (co)homology of
, as shown
here. The cohomology
is a polynomial ring with generators the
Pontryagin classes, so that Pontryagin numbers give an additive isomorphism
.
Since all products of
have linearly independent collections of Pontryagin numbers,
there is a ring isomorphism from
to a polynomial ring with generators
.
Averbuch [Averbuh1959], Milnor [Milnor1960], Thom showed that has no odd torsion and Novikov [Novikov1960] showed that
is isomorphic to a polynomial ring
. Here a generator
can be any
-dimensional manifold such that the Pontryagin number
equals
, if
is not a prime power, or equals
, if
is a power of the prime
.
(Here
is the polynomial which expresses
in terms of the elementary symmetric polynomials of the
.)
Wall [Wall1960], using earlier results of [Rokhlin1953], determined the structure of completely. In particular he proved the following theorems.
Theorem 3.1 [Wall1960, Theorem 2].
All torsion in is of exponent 2.
Theorem 3.2 [Wall1960, Corollary 1].
Two closed oriented n-manifolds and
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.](/images/math/2/c/d/2cd5093093fb6f6b11878497c21a829f.png)
For the complete ring structure, we first describe the subalgebra of the unoriented bordism ring
consisting of classes which contain a manifold
whose first Stiefel-Whitney class is the reduction of an integral class.
is a polynomial ring with coefficients
on generators
where neither
nor
are powers of 2, together with generators
where
is a power of 2. These generators can be described explicitly as follows:
- For
with integers
and
(i.e.
not a power of 2), the generator
is the Dold manifold
and the generator
is the mapping torus of the map
given by the reflection of
at the equator.
- For
a power of 2, the generator
is
. This generator is also represented by
.
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](/images/math/3/1/a/31a5700a7ee612b9086362a9b3b74c3a.png)
where the ring homomorphism is induced by the forgetful map
, and
is the derivation
.
Together with the result that one can choose generators for
such that
,
this determines the ring structure of
.
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
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