Spin bordism

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Contents

1 Introduction

By the Pontrjagin-Thom isomorphism the spin bordism groups \Omega_n^{Spin} of closed manifolds with spin structures are isomorphic to the homotopy groups of the Thom spectrum MSpin.

Preliminary results were by Novikov. The main calculation was achieved in [Anderson&Brown&Peterson1966] and [Anderson&Brown&Peterson1967] where it is shown that all torsion is of exponent 2, being of two types: that arising by products with a framed S^1 and that which maps monomorphically into unoriented cobordism.

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.

For a multi-index 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. By the theorem of Anderson, Brown and Peterson below 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,
  • Z_i

such that the characteristic numbers \pi^J(M_J), \pi^J(N_J) and z_i(Z_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). If all j_i are even, one can choose M_J to be a product of quaternionic projective spaces.

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,
  • Z_i.

3 Invariants

The spin bordism class of a manifold is detected by \Zz_2-cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers.

A spin structure induces a KO-orientation [M], so that we can evaluate polynomials in the KO-Pontryagin classes \pi^j to get characteristic numbers
\displaystyle \pi^J(M)=\langle \pi^J(TM), [M] \rangle \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].

4 Classification

After inverting 2 the map of Thom spectra MSpin\to MSO becomes a homotopy equivalence. Thus there is no odd torsion in the spin cobordism groups, and all \Zz summands are in degrees divisible by 4.

Using the Thom isomorphism KO(BSpin)\cong KO(MSpin) we get for each multi-index J 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] 4.1. There are classes z_i\in H^*(MSpin;\Zz_2) such that there is a 2-local homotopy equivalence

\displaystyle (\pi^J,z_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^{|z_i|}H\Zz_2.

From this one can compute the additive structure completely. Concerning the multiplicative structure, \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 the ideal consisting of torsion mapping monomorphically into unoriented cobordism. According to [Laures2003], the multiplicative structure of this ideal is still not completely known.

5 Further topics

Relation to framed bordism: The image of framed bordism 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 bordism is all classes for which the characteristic numbers divisible by w_1 and w_2 are zero.

6 References

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

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