Spin bordism

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(KO-Characteristic numbers)
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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 $MSpin$.
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{{cite|Stong1968}} {{cite|Laures2003}}
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Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson
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{{cite|Anderson&Brown&Peterson1966}}
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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>
== Calculation ==
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== Invariants ==
<wikitex>;
<wikitex>;
{{cite|Stong1968}}
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The spin bordism class of a manifold is detected by $\Zz_2$-cohomology (Stiefel-Whitney) and
Preliminary results were by Novikov. The main
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KO-theory (Pontryagin) characteristic numbers.
calculation is due to Anderson, Brown, and Peterson
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(Since a spin structure induces a KO-orientation, we can evaluate
{{cite|Anderson&Brown&Peterson1966}}
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polynomials in the [[KO-Characteristic classes|KO-Pontryagin classes]]
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.
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$\pi^j$ to get characteristic numbers in $KO_n(pt)$.)
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These characteristic numbers can be defined as the indices of
Characteristic numbers: Cobordism is determined by $\Zz_2$ cohomology and $KO$-theory characteristic numbers.
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Clifford-linear Dirac operators twisted with the corresponding vector bundles. See
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{{cite|Lawson&Michelsohn1989}}.
Relation to framed bordism: The image of framed cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$.
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Relation to oriented bordism:
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$MSpin\to MSO$ is an equivalence after inverting 2.
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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.
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The image in unoriented cobordism is all classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero.
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</wikitex>
</wikitex>
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== Classification ==
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<wikitex>;
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$MSpin\to MSO$ is an equivalence after inverting 2. Thus there is no odd torsion in the spin cobordism groups.
== KO-Characteristic numbers ==
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For $J=(j_1, \dots, j_n)$ where all $j_i>1$, we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$.
<wikitex>;
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Since Spin-manifolds have a natural KO-orientation, we can evaluate
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polynomials in the [[KO-Characteristic classes|KO-Pontryagin classes]] $\pi^j$ to get characteristic numbers.
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For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$.
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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.
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and $ko\langle 4n(J)-2\rangle$ if $n(J)$ is odd. Hence the corresponding characteristic numbers vanish
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for manifolds of smaller dimension.
{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}}
{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}}
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{{endthm}}
{{endthm}}
</wikitex>
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== Generators ==
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== Low dimensions ==
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<wikitex>;
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$\Omega_0^{Spin}=\Zz$, generated by a point.
$\Omega_0^{Spin}=\Zz$, generated by a point.
$\Omega_1^{Spin}=\Zz_2$, generated by the circle
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$\Omega_1^{Spin}=\Zz_2$, generated by $\alpha$, the circle
with the "antiperiodic" spin structure.
with the "antiperiodic" spin structure.
$\Omega_2^{Spin}=\Zz_2$, generated by the square of the generator of the first bordism group.
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$\Omega_2^{Spin}=\Zz_2$, generated by $\alpha^2$.
$\Omega_3^{Spin}=0$.
$\Omega_3^{Spin}=0$.
$\Omega_4^{Spin}=\Zz$, generated by the Kummer surface.
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$\Omega_4^{Spin}=\Zz$, 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.
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$\Omega_8^{Spin}=\Zz\oplus \Zz$, generated by quaternionic projective space,
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and a generator $\omega$ which equals 1/4 of the square of the Kummer surface.
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By the above theorem of Anderson, Brown and Peterson there exist manifolds
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* $M_J$ of dimension $4n(J)$ if $n(J)$ is even,
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* $N_J$ of dimension $4n(J)-2$ if $n(J)$ is odd,
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* $X_i$
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such that the characteristic numbers $\pi^J(M_J)$, $\pi^J(N_J)$ and $x_i(X_i)$ are odd.
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For $n(J)$ odd, let $W_J$ be a spin nullbordism of $N_J\times \alpha$, and let
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$M_J=W_J\cup W_J$ (using an orientation-reversing automorphism of $\alpha$).
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Then a basis for $\Omega_*^{Spin}\otimes \Qq$ is given by
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* $M_J\times \omega^k$
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* $M_J\times \tau \times \omega^k$.
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A basis for $\Omega_*^{Spin}\otimes \Zz_2$ is given by
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* $M_J\times \omega^k \times \alpha^i$ with $i\le 2$ and $n(J)$ even,
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* $M_J\times \omega^k \times \tau$ with $n(J)$ even,
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* $M_J\times \omega^k $ with $n(J)$ odd,
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* $\frac 14 M_J \times \omega^k \times \alpha^i \times \tau $ with $i\le 2$ and $n(J)$ odd
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* $N_J$,
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* $X_i$.
</wikitex>
</wikitex>
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= Further topics =
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Relation to framed bordism: The image of framed cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$.
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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.
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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 \Omega_n^{Spin} of manifolds with spin structures are the homotopy groups of the Thom spectrum MSpin.

[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 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.

2 Invariants

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-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 [Lawson&Michelsohn1989].

3 Classification

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. 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 and ko\langle 4n(J)-2\rangle if n(J) is odd. Hence the corresponding characteristic numbers vanish for manifolds of smaller dimension.

Theorem [Anderson&Brown&Peterson1966] 3.1. There are classes x_i\in H^*(MSpin;\Zz_2) such that there is a 2-local homotopy equivalence

\displaystyle (\pi^J,x_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle  \vee \bigvee_{n(J)odd, \\ 1\not \in J}ko \langle 4n(J)-2 \rangle  \vee \bigvee_{i}\Sigma^{|x_i|}H\Zz_2

Generators

\Omega_0^{Spin}=\Zz, generated by a point.

\Omega_1^{Spin}=\Zz_2, generated by \alpha, the circle with the "antiperiodic" spin structure.

\Omega_2^{Spin}=\Zz_2, generated by \alpha^2.

\Omega_3^{Spin}=0.

\Omega_4^{Spin}=\Zz, generated by \tau, the Kummer surface.

\Omega_5^{Spin}=\Omega_6^{Spin}=\Omega_7^{Spin}=0.

\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.

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

This page has not been refereed. The information given here might be incomplete or provisional.

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