Spin bordism

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== KO-Characteristic classes ==
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== KO-Characteristic numbers ==
<wikitex>;
<wikitex>;
%This should probably get its own page.%
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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.
The KO-Pontryagin classes $\pi^j$ are defined by setting
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For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$.
$\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$
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Using the Thom isomorphism $KO(BSpin)\cong KO(MSpin)$ we get a map $\pi^J:MSpin\to KO$.
for complex line bundles L and then requiring naturality and
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$\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$
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where $\pi_s =\sum_j \pi^j s^j$ .
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Here $\xi$ and $\eta$ are oriented bundles.
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In fact, these properties determine $\pi^j$ because the group $KO(BSO(m))$ injects into $K(BT^{[m/2]})$ under the
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complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$
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(compare {{cite|Anderson&Brown&Peterson1966}}).
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For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$.
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Such a class gives a map $MSpin\to ko \langle m\rangle$.
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{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}}
{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}}
There is a 2-local homotopy equivalence
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There are classes $x_i\in H^*(MSpin;\Zz_2)$ such that there is a 2-local homotopy equivalence
$$MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle
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$$(\pi^J,x_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle
\vee
\vee
\bigvee_{n(J)odd, \\ 1\not \in J}ko \langle 4n(J)+2 \rangle
\bigvee_{n(J)odd, \\ 1\not \in J}ko \langle 4n(J)+2 \rangle

Revision as of 13:12, 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.


2 Calculation

[Stong1968] 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.

Characteristic numbers: Cobordism is determined by \Zz_2 cohomology and KO-theory characteristic numbers.

Relation to framed bordism: The image of framed cobordism is 0 except in dimensions 8k+1,8k+2 where it is \Zz_2.

Relation to oriented bordism: MSpin\to MSO is an equivalence after inverting 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.


3 KO-Characteristic numbers

Since Spin-manifolds have a natural KO-orientation, we can evaluate polynomials in the KO-Pontryagin classes \pi^j to get characteristic numbers.

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. Using the Thom isomorphism KO(BSpin)\cong KO(MSpin) we get a map \pi^J:MSpin\to KO.

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



4 Low dimensions

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

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

\Omega_2^{Spin}=\Zz_2, generated by the square of the generator of the first bordism group.

\Omega_3^{Spin}=0.

\Omega_4^{Spin}=\Zz, generated by 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 1/4 of the square of the Kummer surface.

5 References

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

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