Spin bordism

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=== MSpin away from the prime 2 and at the prime 2 ===
=== MSpin away from the prime 2 and at the prime 2 ===
<wikitex>;
<wikitex>;
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.
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After inverting 2 the map of Thom spectra $MSpin\to MSO$ becomes a homotopy equivalence.
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
+
Using the Thom isomorphism $KO(BSpin)\cong KO(MSpin)$ we get for each multi-index $J$ a map $\pi^J:MSpin\to KO$,
Anderson,Brown and Peterson show that it factorizes through $ko\langle 4n(J)\rangle$ if $n(J)$ is even
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which induces on homotopy groups the map $\pi_*(MSpin)\to \pi_*KO$, $[M]\mapsto \pi^J(M)$ described above.
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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
and $ko\langle 4n(J)-2\rangle$ if $n(J)$ is odd. Hence the corresponding characteristic numbers vanish
for manifolds of smaller dimension.
for manifolds of smaller dimension.
{{beginthm|Theorem|{{cite|Anderson&Brown&Peterson1967}}}}
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Similarly a Stiefel-Whitney class in $H^j(BSpin;\Zz_2)\cong H^j(MSpin;\Zz_2)$ corresponds to a spectrum map $MSpin\to \Sigma^{j}H\Zz_2$.
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{{beginthm|Theorem|{{cite|Anderson&Brown&Peterson1967}}}} \label{ABPthm}
There are classes $z_i\in H^*(MSpin;\Zz_2)$ such that there is a 2-local homotopy equivalence
There are classes $z_i\in H^*(MSpin;\Zz_2)$ such that there is a 2-local homotopy equivalence
$$(\pi^J,z_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle
$$(\pi^J,z_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle
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{{endthm}}
{{endthm}}
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{{cite|Anderson&Brown&Peterson1966}} also determine the Poincaré polynomial of $H^*(MSpin;\Zz_2)$ which allows to compute inductively
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the degrees $|z_i|$. The first $z_i$ occurs in dimension $20$, and there are $z_i$ in all dimensions $\ge 36$.
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=== Consequences ===
=== Consequences ===
<wikitex>;
<wikitex>;
From this one can compute the additive structure completely.
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From this one can compute the additive structure of the spin bordism groups completely.
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.
+
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
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$\Zz,\Zz_2,\Zz_2,0,\Zz,0,0,0,\Zz$ (here the first value corresponds to dimensions congruent to 0 modulo 8).
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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.
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=== Ring structure ===
=== Ring structure ===
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$\Omega_8^{Spin}=\Zz\oplus \Zz$, generated by [[Wikipedia:Quaternionic_projective_space#Quaternionic_projective_plane|quaternionic projective space]] and a generator $\omega_8$ which equals 1/4 of the square of the Kummer surface.
$\Omega_8^{Spin}=\Zz\oplus \Zz$, generated by [[Wikipedia:Quaternionic_projective_space#Quaternionic_projective_plane|quaternionic projective space]] and a generator $\omega_8$ which equals 1/4 of the square of the Kummer surface.
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=== Generators in all dimensions as given by the classification ===
=== Generators in all dimensions as given by the classification ===
<wikitex>;
<wikitex>;
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 $n(J)=\sum_i j_i$.
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
By the theorem of Anderson, Brown and Peterson below there exist manifolds
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* $Z_i$
* $Z_i$
such that the characteristic numbers $\pi^J(M_J)$, $\pi^J(N_J)$ and $z_i(Z_i)$ are odd.
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
+
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 {{cite|Stong1966}} there exist manifolds $P_J$ such that
$M_J=W_J\cup W_J$ (using an orientation-reversing automorphism of $\alpha$). By {{cite|Stong1966}} there exist manifolds $P_J$ such that
$4 P_J = M_J\times \tau$. If all $j_i$ are even,
+
$4 P_J = M_J\times \tau_4$. If all $j_i$ are even,
one can choose $M_J$ to be a product of quaternionic projective spaces.
one can choose $M_J$ to be a product of quaternionic projective spaces.
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=== Relationship with other bordism groups ===
=== Relationship with other bordism groups ===
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* [[Framed bordism|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 $\omega^k\times \alpha^i$ with $i\le 2$ and $k\ge 0$.
+
* [[Framed bordism|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|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 $M_J\times \omega_8^i \times\alpha_1^i$, $n(J)$ even, and $P_J\times \omega_8^i \times\alpha_1^i$,$n(J)$ 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 $\Omega^{SO}_8/\Omega^{Spin}_8\cong \Zz_{2^7}$.
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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 $\Zz_2$ 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 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 $\ge 36$.
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* [[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 $M_J$, 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.
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Revision as of 16:58, 22 September 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].


2 Invariants

The spin bordism class of a manifold is detected by
Tex syntax error
-cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers. 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 possibly a factor of 2) the \hat{A}-genus of M.

3 Classification

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(BSpin)\cong KO(MSpin) we get for each multi-index 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_{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.

[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.

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
Tex syntax error
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.

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.

4 Generators

4.1 Low dimensions

\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_3^{Spin}=0.

\Omega_4^{Spin}=\Zz, generated by \tau_4, 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_8 which equals 1/4 of the square of the Kummer surface.

4.2 Generators in all dimensions as given by the classification

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_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
  • M_J\times \tau_4 \times \omega_8^k.

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

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


5 Further topics

5.1 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
    Tex syntax error
    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
    Tex syntax error
    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 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, 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.

6 References

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