Oriented bordism
Contents |
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 space
.
, generated by the Wu manifold.
.
is a polynomial ring, with generators
.
3 Invariants
Signature. Pontryagin numbers. Stiefel-Whitney numbers.
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, Milnor [Milnor1960], Thom showed that has no odd torsion and
is isomorphic to a polynomial ring
. Here the generators
can be any
-dimensional manifolds 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 Rohlin, determined the structure of completely.
In particular he proved that all torsion in
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 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 on the following generators.
- For
with integers
and
(i.e.
not a power of 2), we have generators
, the Dold manifolds.
- Reflection of
at the equator induces a map
. The generator
is the mapping torus of this map.
- For
a power of 2, the generator
.
Now there is an exact sequence
![\displaystyle \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](/images/math/7/b/c/7bcacb88e6ba1f4be07204a4711ce0af.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 Further topics
6 References
- [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
- [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
This page has not been refereed. The information given here might be incomplete or provisional. |