Spin bordism

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* [[Oriented bordism|Oriented bordism]]: the kernel of $\Omega_*^{Spin} \to \Omega_*^{SO}$ lies in dimensions $8k + 1$ and $8k + 2$.It is a $\Zz$ vector space with a basis $M_J\times \omega^i \times\alpha^i$, $n(J)$ even, and $P_J\times \omega^i \times\alpha^i$,$n(J)$ odd where $k\ge 0$ and $i\le 2$. It is also the ideal generated by the non-trivial class of $\Omega_1^{Spin} = \Zz/2$. The cokernel is a finite $2$-torsion group which is trivial if and only of $\Omega^{SO}_\ast = 0$ or equivalently $\ast=1$, $2$, $3$, $6$, and $7$. {{cite|Milnor1965}} computed $\Omega^{SO}_8/\Omega^{Spin}_8\cong \Zz_{2^7}$.
* [[Oriented bordism|Oriented bordism]]: the kernel of $\Omega_*^{Spin} \to \Omega_*^{SO}$ lies in dimensions $8k + 1$ and $8k + 2$.It is a $\Zz$ vector space with a basis $M_J\times \omega^i \times\alpha^i$, $n(J)$ even, and $P_J\times \omega^i \times\alpha^i$,$n(J)$ odd where $k\ge 0$ and $i\le 2$. It is also the ideal generated by the non-trivial class of $\Omega_1^{Spin} = \Zz/2$. The cokernel is a finite $2$-torsion group which is trivial if and only of $\Omega^{SO}_\ast = 0$ or equivalently $\ast=1$, $2$, $3$, $6$, and $7$. {{cite|Milnor1965}} computed $\Omega^{SO}_8/\Omega^{Spin}_8\cong \Zz_{2^7}$.
* [[Unoriented bordism|Unoriented bordism]]: the image of $\Omega_*^{Spin} \to \mathcal{N}_*$ is all bordism classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero. A basis for the image consists of the $M_J$, $n(J)$ even, the $N_J$, $n(J)$ odd and the $Z_i$. The first $Z_i$ occurs in dimension $20$. The image is trivial for $\ast<8$. In even dimensions it is additionally trivial for $\ast=14$ and $16$. In odd dimensions it is trivial for $\ast<29$ and also for $\ast=31$ and $35$. Otherwise the image is non trivial. Indeed, there are $Z_i$ in all dimensions $\ge 36$.
* [[Unoriented bordism|Unoriented bordism]]: the image of $\Omega_*^{Spin} \to \mathcal{N}_*$ is all bordism classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero. A basis for the image consists of the $M_J$, $n(J)$ even, the $N_J$, $n(J)$ odd and the $Z_i$. The first $Z_i$ occurs in dimension $20$. The image is trivial for $\ast<8$. In even dimensions it is additionally trivial for $\ast=14$ and $16$. In odd dimensions it is trivial for $\ast<29$ and also for $\ast=31$ and $35$. Otherwise the image is non trivial. Indeed, there are $Z_i$ in all dimensions $\ge 36$.

Revision as of 17:47, 10 July 2010

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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). By [Stong1966] there exist manifolds P_J such that 4 P_J = M_J\times \tau. 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,
  • P_J \times \omega^k \times \alpha^i 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_*(pt).

A feature of Spin manifolds M is that they possess Dirac operators, \partial\!\!\!/_M. The (Clifford-linear) Dirac operator can be considered as a representative of the fundamental class [M]\in KO_n(M), where n is the dimension of M, see [Atiyah1970] and [Higson&Roe2000]. The characteristic numbers above can then be defined as the indices of the (Clifford-linear) Dirac operators obtained by twisting with the corresponding vector bundles. The easiest case is the non-twisted one: \pi^0(TM) is the trivial bundle, and taking the index of the Dirac operator \partial\!\!\!/_M defines an element of KO^{-n}(pt)\cong KO_n(pt) when M is n-dimensional. This gives rises to a ring homomorphism often called the \alpha-invariant:

\displaystyle  \alpha : \Omega_*^{Spin} \longrightarrow KO_*, ~~~ [M] \longmapsto [\operatorname{ind} \partial\!\!\!/_M]=\alpha(M)=\pi^0(M).

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

5.1 Relationship with other bordism groups

  • Framed bordism the image of \Omega_*^{fr} \to \Omega_*^{Spin} is 0 except in unless * = 8k+1 or 8k+2 when it is \Zz_2 and detected by the \alpha-invariant. Explicit generators are \omega^k\times \alpha^i with i\le 2 and k\ge 0.
  • Oriented bordism: the kernel of \Omega_*^{Spin} \to \Omega_*^{SO} lies in dimensions 8k + 1 and 8k + 2.It is a \Zz vector space with a basis M_J\times \omega^i \times\alpha^i, n(J) even, and P_J\times \omega^i \times\alpha^i,n(J) odd where k\ge 0 and i\le 2. It is also the ideal generated by the non-trivial class of \Omega_1^{Spin} = \Zz/2. The cokernel is a finite 2-torsion group which is trivial if and only of \Omega^{SO}_\ast = 0 or equivalently \ast=1, 2, 3, 6, and 7. [Milnor1965] computed \Omega^{SO}_8/\Omega^{Spin}_8\cong \Zz_{2^7}.
  • Unoriented bordism: the image of \Omega_*^{Spin} \to \mathcal{N}_* is all bordism classes for which the characteristic numbers divisible by w_1 and w_2 are zero. A basis for the image consists of the M_J, n(J) even, the N_J, n(J) odd and the Z_i. The first Z_i occurs in dimension 20. The image is trivial for \ast<8. In even dimensions it is additionally trivial for \ast=14 and 16. In odd dimensions it is trivial for \ast<29 and also for \ast=31 and 35. Otherwise the image is non trivial. Indeed, there are Z_i in all dimensions \ge 36.

6 References

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