Spin bordism

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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

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 classes

%This should probably get its own page.%

The KO-Pontryagin classes \pi^j are defined by setting \pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0 for j \ge 2 for complex line bundles L and then requiring naturality and \pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta) where \pi_s =\sum_j \pi^j s^j . Here \xi and \eta are oriented bundles. In fact, these properties determine \pi^j because the group KO(BSO(m)) injects into K(BT^{[m/2]}) under the complexification of the map which is induced by the restriction to the maximal torus T^{[m/2]} (compare [Anderson&Brown&Peterson1966]). For J=(j_1,\dots,  j_n) we set \pi^J=\pi^{j_1}\dots \pi^{j_n}. Such a class gives a map MSpin\to ko \langle m\rangle.

Theorem [Anderson&Brown&Peterson1966] 3.1. There is a 2-local homotopy equivalence

\displaystyle 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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