Spin bordism

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

Revision as of 09:39, 28 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.

Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson [Anderson&Brown&Peterson1967] 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.

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&Peterson1967] 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

Stong proves that \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 [Stong1968]. Anderson, Brown and Peterson determine the structure of \Omega_*^{Spin} modulo torsion mapping monomorphically into unoriented cobordism.

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.

5 References

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

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