Oriented bordism
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(Here $s_i$ is the polynomial which expresses $\sum t_k^i$ in terms of the elementary symmetric polynomials of the $t_i$.) | (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 Rohlin, determined the structure of $\Omega_*^{SO}$ completely. | + | Wall {{cite|Wall1960}}, using earlier results of Rohlin, determined the structure of $\Omega_*^{SO}$ completely. In particular he proved the following theorems. |
− | In particular he proved | + | |
− | + | ||
− | For the complete structure, we first describe the subalgebra $\mathcal{W}$ of the unoriented bordism ring $\mathcal{N}$ | + | {{beginthm|Theorem|Cf. {{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. | 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 on the following generators. | $\mathcal{W}$ is a polynomial ring on the following generators. | ||
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Now there is an exact sequence | Now there is an exact sequence | ||
− | $$ \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to $$ | + | $$ \dots \to \Omega_q^{SO} \stackrel 2 \to \Omega_q^{SO} \stackrel r \to \mathcal{W}_q \stackrel \partial \to \Omega_{q-1}^{SO} \stackrel 2 \to \Omega_{q-1}^{SO} \to \dots $$ |
where the ring homomorphism $r$ is induced by the forgetful map $\Omega_q^{SO} \to \mathcal{N}$, and $r\partial:\mathcal{W}\to \mathcal{W}$ | 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 $X_{2k}\mapsto X_{2k-1}, X_{2k-1}\mapsto 0, X_k^2\mapsto 0$. | is the derivation $X_{2k}\mapsto X_{2k-1}, X_{2k-1}\mapsto 0, X_k^2\mapsto 0$. |
Revision as of 16:53, 4 February 2010
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 plane .
- , generated by the Wu manifold .
- .
- generated by and .
is a polynomial ring, with generators .
3 Invariants
The signature of a closed oriented manifold is a fundamental bordism invariant defining a ring homomorphism
(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
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&Stasheff(1974), Theorm 4.9, Lemma 17.3]) and clearly additive. Hence obtain homomorphisms
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 a
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 the following theorems.
Theorem 4.1 Cf. [Wall1960, Theorem 2]. All torsion in is of exponent 2.
Theorem 4.2 [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:
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 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
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
- [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&Stasheff(1974)] Template:Milnor&Stasheff(1974)
- [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
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