Oriented bordism
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== Invariants == | == Invariants == | ||
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− | [[Wikipedia:Pontryagin class#Pontryagin numbers|Pontryagin | + | The [[Wikipedia:Signature_(topology)|signature]] of a closed oriented manifold is a fundamental bordism invariant defining a ring homomorphism |
− | [[Wikipedia:Stiefel-Whitney class#Stiefel–Whitney numbers|Stiefel-Whitney numbers]]. | + | $$ \sigma : \Omega_*^{SO} \to \Zz.$$ |
+ | (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 | ||
+ | [[Wikipedia:Pontryagin class#Pontryagin numbers|Pontryagin number]] $p_J$ of a closed, oriented manifold $M$ of dimension $4n(J)$ is the integer | ||
+ | $$ \langle p_{j_1}(M) \cup p_{j_2}(M) \cup \dots \cup p_{j_n}(M), [M]\rangle \in \Zz $$ | ||
+ | where $p_{k}$ is the k-the Pontrjagin of $M$ and $[M]$ its [[Wikipedia:Fundamental_class|fundamental class]]. The [[Wikipedia:Stiefel-Whitney class#Stiefel–Whitney numbers|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 {{cite|Milnor&Stasheff(1974)|Theorm 4.9, Lemma 17.3}}) and clearly additive. Hence obtain homomorphisms | ||
+ | $$ p_J : \Omega_{n(J)}^{SO} \to \Zz \text{~~and~~} w_J : \Omega_{n(J)}^{SO} \to \Zz/2.$$ | ||
+ | By Hirzebruch's [[Wikipedia:Genus_of_a_multiplicative_sequence#L_genus_and_the_Hirzebruch_signature_theorem|signature theorem]] {{cite|Hirzebruch1953}}, {{cite|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 a$M$ | ||
+ | $$ \sigma(M) = \langle L(p_1(M), \dots , p_n(M)), [M] \rangle.$$ | ||
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Revision as of 15:37, 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 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
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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