Spin bordism

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[edit] 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].

[edit] 2 Invariants

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

For a multi-index J=(j_1, \dots, j_n), we set \pi^J=\pi^{j_1}\dots \pi^{j_n} and n(J)=\sum_i j_i.

A spin structure on a closed n-manifold M induces a KO-orientation [M]\in KO_n(M), so that we can evaluate polynomials in the KO-Pontryagin classes \pi^j(TM)\in KO^0(M) to get characteristic numbers
\displaystyle \pi^J(M)=\langle \pi^J(TM), [M] \rangle \in KO_n(pt).

There is an interpretation of these characteristic numbers using index theory:

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), 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]. For n divisible by 4, this invariant equals (up to a factor of 1/2 in dimensions congruent to 4 modulo 8) the \hat{A}-genus of M.

[edit] 3 Classification

[edit] 3.1 MSpin away from the prime 2 and at the prime 2

After inverting 2 the map of Thom spectra MSpin\to MSO becomes a homotopy equivalence.

Using the Thom isomorphism KO^0(BSpin)\cong KO^0(MSpin) we get for each multi-index J with 1\not \in J a map \pi^J:MSpin\to KO, which induces on homotopy groups the map \pi_*(MSpin)\to \pi_*KO, [M]\mapsto \pi^J(M) described above. Anderson,Brown and Peterson show that \pi^J 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.

Similarly a Stiefel-Whitney class in H^j(BSpin;\Zz_2)\cong H^j(MSpin;\Zz_2) corresponds to a spectrum map MSpin\to \Sigma^{j}H\Zz_2.

Theorem 3.1 [Anderson&Brown&Peterson1967]. 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_{\stackrel{n(J) \text{even,}}{1\not \in J}}  ko \langle 4n(J)\rangle  \vee\bigvee_{\stackrel{n(J)\text{odd,}} {1\not \in J}}ko \langle 4n(J)-2 \rangle \vee\bigvee_{i}\Sigma^{|z_i|}H\Zz_2.

[Anderson&Brown&Peterson1966] also determine the Poincaré polynomial of H^*(MSpin;\Zz_2) which allows to compute inductively the degrees |z_i|. The first z_i occurs in dimension 20, and there are z_i in all dimensions \ge 36.

[edit] 3.2 Consequences

From this one can compute the additive structure of the spin bordism groups completely. We get a contribution from each ko\langle m\rangle which is 0 below dimension m, and periodic of period 8 starting from dimension m, with values \Zz,\Zz_2,\Zz_2,0,\Zz,0,0,0,\Zz (here the first value corresponds to dimensions congruent to 0 modulo 8). The contribution from \Sigma^{|z_i|}H\Zz_2 is a single \Zz_2 in dimension |z_i|.

All \Zz summands are in degrees divisible by 4, and there is no odd torsion in the spin cobordism groups. All even 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.

[edit] 3.3 Ring structure

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.

[edit] 4 Generators

[edit] 4.1 Low dimensions

The spin bordism groups up to dimension 8 are given in [Milnor1963a] without proof. Milnor states that this is the result of a formibable calculation of \pi_i(MSpin) for i \leq 8.

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

\Omega_1^{Spin}=\Zz_2, generated by \alpha_1, the circle with the "antiperiodic" spin structure.

\Omega_2^{Spin}=\Zz_2, generated by \alpha_1^2.


\Omega_4^{Spin}=\Zz, generated by \tau_4, the Kummer surface.


\Omega_8^{Spin}=\Zz\oplus \Zz, generated by quaternionic projective space and a generator \omega_8 which equals 1/4 of the square of the Kummer surface.

[edit] 4.2 Generators in all dimensions as given by the classification

By the 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, and Z_i of dimension |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_1, 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_4. 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_8^k for k\ge 0
  • M_J\times \tau_4 \times \omega_8^k for k\ge 0.

A basis for \Omega_*^{Spin}\otimes \Zz_2 is given by

  • M_J\times \omega_8^k \times \alpha_1^i with k\ge 0, i\le 2 and n(J) even,
  • M_J\times \omega_8^k \times \tau_4 with k\ge 0, n(J) even,
  • M_J\times \omega_8^k with k\ge 0, n(J) odd,
  • P_J \times \omega_8^k \times \alpha_1^i with k\ge 0, i\le 2 and n(J) odd
  • N_J with n(J) odd,
  • Z_i.

[edit] 5 Further topics

[edit] 5.1 Rohlin's theorem

Above we stated that the 4-dimensional spin bordism group \Omega_4^{Spin} \cong \Z is generated by the Kummer surface K3 which has signature 16. Consequently we have the following important theorem of Rokhlin (which of course was used in calculation of \Omega_2^{Spin} give above).

Theorem 5.1 [Rohlin1952]. The signature of every closed smooth spin 4-manifold is divisible by 16.

[edit] 5.2 Relationship with other bordism groups

  • Framed bordism: the image of \Omega_*^{fr} \to \Omega_*^{Spin} is 0 unless * = 8k+1 or 8k+2 when it is \Zz_2 and detected by the \alpha-invariant. Explicit generators are \omega_8^k\times \alpha_1^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_2 vector space with a basis M_J\times \omega_8^i \times\alpha_1^i, for n(J) even, and P_J\times \omega_8^i \times\alpha_1^i, for 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 if \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, for n(J) even, the N_J, for n(J) odd and the Z_i. 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.

[edit] 6 References

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