Spin bordism
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The spin bordism groups $\Omega_n^{Spin}$ of manifolds with spin structures are the homotopy groups of the Thom spectrum $MSpin$. | The spin bordism groups $\Omega_n^{Spin}$ of manifolds with spin structures are the homotopy groups of the Thom spectrum $MSpin$. | ||
+ | {{cite|Stong1968}} {{cite|Laures2003}} | ||
+ | Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson | ||
+ | {{cite|Anderson&Brown&Peterson1966}} | ||
+ | who showed that all torsion is of order 2, being of two types: that arising by products with a framed $S^1$ and that which maps monomorphically into unoriented cobordism. $\Omega_*^{Spin}/Torsion$ is the subring of an integral polynomial ring on classes $x_{4i}$ (dimension 4i) consisting of all classes of dimension a multiple of 8 and twice the classes whose dimension is not a multiple of 8. | ||
</wikitex> | </wikitex> | ||
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+ | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | + | The spin bordism class of a manifold is detected by $\Zz_2$-cohomology (Stiefel-Whitney) and | |
− | + | KO-theory (Pontryagin) characteristic numbers. | |
− | + | (Since a spin structure induces a KO-orientation, we can evaluate | |
− | + | polynomials in the [[KO-Characteristic classes|KO-Pontryagin classes]] | |
− | + | $\pi^j$ to get characteristic numbers in $KO_n(pt)$.) | |
− | + | These characteristic numbers can be defined as the indices of | |
− | + | Clifford-linear Dirac operators twisted with the corresponding vector bundles. See | |
− | + | {{cite|Lawson&Michelsohn1989}}. | |
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− | $ | + | |
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</wikitex> | </wikitex> | ||
+ | == Classification == | ||
+ | <wikitex>; | ||
+ | $MSpin\to MSO$ is an equivalence after inverting 2. Thus there is no odd torsion in the spin cobordism groups. | ||
− | + | For $J=(j_1, \dots, j_n)$ where all $j_i>1$, we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$. | |
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− | For $J=(j_1,\dots, | + | |
Using the Thom isomorphism $KO(BSpin)\cong KO(MSpin)$ we get a map $\pi^J:MSpin\to KO$, for which | Using the Thom isomorphism $KO(BSpin)\cong KO(MSpin)$ we get a map $\pi^J:MSpin\to KO$, for which | ||
Anderson,Brown and Peterson show that it factorizes through $ko\langle 4n(J)\rangle$ if $n(J)$ is even | Anderson,Brown and Peterson show that it factorizes through $ko\langle 4n(J)\rangle$ if $n(J)$ is even | ||
− | and $ko\langle 4n(J)-2\rangle$ if $n(J)$ is odd. | + | and $ko\langle 4n(J)-2\rangle$ if $n(J)$ is odd. Hence the corresponding characteristic numbers vanish |
+ | for manifolds of smaller dimension. | ||
{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}} | {{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}} | ||
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{{endthm}} | {{endthm}} | ||
− | + | == Generators == | |
− | + | ||
− | == | + | |
− | + | ||
$\Omega_0^{Spin}=\Zz$, generated by a point. | $\Omega_0^{Spin}=\Zz$, generated by a point. | ||
− | $\Omega_1^{Spin}=\Zz_2$, generated by the circle | + | $\Omega_1^{Spin}=\Zz_2$, generated by $\alpha$, the circle |
with the "antiperiodic" spin structure. | with the "antiperiodic" spin structure. | ||
− | $\Omega_2^{Spin}=\Zz_2$, generated by | + | $\Omega_2^{Spin}=\Zz_2$, generated by $\alpha^2$. |
$\Omega_3^{Spin}=0$. | $\Omega_3^{Spin}=0$. | ||
− | $\Omega_4^{Spin}=\Zz$, generated by the Kummer surface. | + | $\Omega_4^{Spin}=\Zz$, generated by $\tau$, the Kummer surface. |
$\Omega_5^{Spin}=\Omega_6^{Spin}=\Omega_7^{Spin}=0$. | $\Omega_5^{Spin}=\Omega_6^{Spin}=\Omega_7^{Spin}=0$. | ||
− | $\Omega_8^{Spin}=\Zz\oplus \Zz$, generated by quaternionic projective space, and 1/4 of the square of the Kummer surface. | + | $\Omega_8^{Spin}=\Zz\oplus \Zz$, generated by quaternionic projective space, |
+ | and a generator $\omega$ which equals 1/4 of the square of the Kummer surface. | ||
+ | |||
+ | By the above theorem of Anderson, Brown and Peterson there exist manifolds | ||
+ | * $M_J$ of dimension $4n(J)$ if $n(J)$ is even, | ||
+ | * $N_J$ of dimension $4n(J)-2$ if $n(J)$ is odd, | ||
+ | * $X_i$ | ||
+ | such that the characteristic numbers $\pi^J(M_J)$, $\pi^J(N_J)$ and $x_i(X_i)$ are odd. | ||
+ | For $n(J)$ odd, let $W_J$ be a spin nullbordism of $N_J\times \alpha$, and let | ||
+ | $M_J=W_J\cup W_J$ (using an orientation-reversing automorphism of $\alpha$). | ||
+ | |||
+ | Then a basis for $\Omega_*^{Spin}\otimes \Qq$ is given by | ||
+ | * $M_J\times \omega^k$ | ||
+ | * $M_J\times \tau \times \omega^k$. | ||
+ | |||
+ | A basis for $\Omega_*^{Spin}\otimes \Zz_2$ is given by | ||
+ | * $M_J\times \omega^k \times \alpha^i$ with $i\le 2$ and $n(J)$ even, | ||
+ | * $M_J\times \omega^k \times \tau$ with $n(J)$ even, | ||
+ | * $M_J\times \omega^k $ with $n(J)$ odd, | ||
+ | * $\frac 14 M_J \times \omega^k \times \alpha^i \times \tau $ with $i\le 2$ and $n(J)$ odd | ||
+ | * $N_J$, | ||
+ | * $X_i$. | ||
</wikitex> | </wikitex> | ||
+ | |||
+ | = Further topics = | ||
+ | |||
+ | Relation to framed bordism: The image of framed cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$. | ||
+ | |||
+ | The kernel of the map from spin to oriented bordism is in dimensions $8k + 1$ and $8k + 2$ only and is the part generated by framed manifolds. It is the ideal generated by the non-trivial class of degree 1. | ||
+ | |||
+ | The image in unoriented cobordism is all classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero. | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
+ | |||
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Revision as of 19:53, 27 January 2010
Contents |
1 Introduction
The spin bordism groups of manifolds with spin structures are the homotopy groups of the Thom spectrum .
[Stong1968] [Laures2003] Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson [Anderson&Brown&Peterson1966] who showed that all torsion is of order 2, being of two types: that arising by products with a framed and that which maps monomorphically into unoriented cobordism. is the subring of an integral polynomial ring on classes (dimension 4i) consisting of all classes of dimension a multiple of 8 and twice the classes whose dimension is not a multiple of 8.
2 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers. (Since a spin structure induces a KO-orientation, we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbers in .) These characteristic numbers can be defined as the indices of Clifford-linear Dirac operators twisted with the corresponding vector bundles. See [Lawson&Michelsohn1989].
3 Classification
is an equivalence after inverting 2. Thus there is no odd torsion in the spin cobordism groups.
For where all , we set and . Using the Thom isomorphism we get a map , for which Anderson,Brown and Peterson show that it factorizes through if is even and if is odd. Hence the corresponding characteristic numbers vanish for manifolds of smaller dimension.
Theorem [Anderson&Brown&Peterson1966] 3.1. There are classes such that there is a 2-local homotopy equivalence
Generators
, generated by a point.
, generated by , the circle with the "antiperiodic" spin structure.
, generated by .
.
, generated by , the Kummer surface.
.
, generated by quaternionic projective space, and a generator which equals 1/4 of the square of the Kummer surface.
By the above theorem of Anderson, Brown and Peterson there exist manifolds
- of dimension if is even,
- of dimension if is odd,
such that the characteristic numbers , and are odd. For odd, let be a spin nullbordism of , and let (using an orientation-reversing automorphism of ).
Then a basis for is given by
- .
A basis for is given by
- with and even,
- with even,
- with odd,
- with and odd
- ,
- .
4 Further topics
Relation to framed bordism: The image of framed cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$.
The kernel of the map from spin to oriented bordism is in dimensions $8k + 1$ and $8k + 2$ only and is the part generated by framed manifolds. It is the ideal generated by the non-trivial class of degree 1.
The image in unoriented cobordism is all classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero.
4.1 References
- [Anderson&Brown&Peterson1966] D. W. Anderson, E. H. Brown and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR0190939 (32 #8349) Zbl 0156.21605
- [Laures2003] G. Laures, An splitting of spin bordism, Amer. J. Math. 125 (2003), no.5, 977–1027. MR2004426 (2004g:55007) Zbl 1058.55001
- [Lawson&Michelsohn1989] H. B. Lawson and M. Michelsohn, Spin geometry, Princeton University Press, Princeton, NJ, 1989. MR1031992 (91g:53001) Zbl 0801.58017
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
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