Spin bordism

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1 Introduction

By the Pontrjagin-Thom isomorphism the spin bordism groups $\Omega_n^{Spin}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Omega_n^{Spin}$ of closed manifolds with spin structures are isomorphic to the homotopy groups of the Thom spectrum $MSpin$$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 $\Zz_2$$\Zz_2$-cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers.

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

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

$\displaystyle \pi^J(M)=\langle \pi^J(TM), [M] \rangle \in KO_n(pt).$ $$\displaystyle \pi^J(M)=\langle \pi^J(TM), [M] \rangle \in KO_n(pt).$$

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

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

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

See [Lawson&Michelsohn1989]. For $n$$n$ divisible by 4, this invariant equals (up to a factor of 1/2 in dimensions congruent to 4 modulo 8) the $\hat{A}$$\hat{A}$-genus of $M$$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$$MSpin\to MSO$ becomes a homotopy equivalence.

Using the Thom isomorphism $KO^0(BSpin)\cong KO^0(MSpin)$$KO^0(BSpin)\cong KO^0(MSpin)$ we get for each multi-index $J$$J$ with $1\not \in J$$1\not \in J$ a map $\pi^J:MSpin\to KO$$\pi^J:MSpin\to KO$, which induces on homotopy groups the map $\pi_*(MSpin)\to \pi_*KO$$\pi_*(MSpin)\to \pi_*KO$, $[M]\mapsto \pi^J(M)$$[M]\mapsto \pi^J(M)$ described above. Anderson,Brown and Peterson show that $\pi^J$$\pi^J$ factorizes through $ko\langle 4n(J)\rangle$$ko\langle 4n(J)\rangle$ if $n(J)$$n(J)$ is even and $ko\langle 4n(J)-2\rangle$$ko\langle 4n(J)-2\rangle$ if $n(J)$$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)$$H^j(BSpin;\Zz_2)\cong H^j(MSpin;\Zz_2)$ corresponds to a spectrum map $MSpin\to \Sigma^{j}H\Zz_2$$MSpin\to \Sigma^{j}H\Zz_2$.

Theorem 3.1 [Anderson&Brown&Peterson1967]. There are classes $z_i\in H^*(MSpin;\Zz_2)$$z_i\in H^*(MSpin;\Zz_2)$ such that there is a 2-local homotopy equivalence

$\displaystyle (\pi^J,z_i): MSpin \to\bigvee_{\stackrel{n(J) \text{even,}}{1\not \in J}} ko \langle 4n(J)\rangle \vee\bigvee_{\stackrel{n(J)\text{odd,}} {1\not \in J}}ko \langle 4n(J)-2 \rangle \vee\bigvee_{i}\Sigma^{|z_i|}H\Zz_2.$ $$\displaystyle (\pi^J,z_i): MSpin \to\bigvee_{\stackrel{n(J) \text{even,}}{1\not \in J}} ko \langle 4n(J)\rangle \vee\bigvee_{\stackrel{n(J)\text{odd,}} {1\not \in J}}ko \langle 4n(J)-2 \rangle \vee\bigvee_{i}\Sigma^{|z_i|}H\Zz_2.$$

[Anderson&Brown&Peterson1966] also determine the Poincaré polynomial of $H^*(MSpin;\Zz_2)$$H^*(MSpin;\Zz_2)$ which allows to compute inductively the degrees $|z_i|$$|z_i|$. The first $z_i$$z_i$ occurs in dimension $20$$20$, and there are $z_i$$z_i$ in all dimensions $\ge 36$$\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$$ko\langle m\rangle$ which is 0 below dimension $m$$m$, and periodic of period 8 starting from dimension $m$$m$, with values $\Zz,\Zz_2,\Zz_2,0,\Zz,0,0,0,\Zz$$\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$$\Sigma^{|z_i|}H\Zz_2$ is a single $\Zz_2$$\Zz_2$ in dimension $|z_i|$$|z_i|$.

All $\Zz$$\Zz$ summands are in degrees divisible by 4, and there is no odd torsion in the spin cobordism groups. All even torsion is of exponent 2, being of two types: that arising by products with a framed $S^1$$S^1$ and that which maps monomorphically into unoriented cobordism.

3.3 Ring structure

Concerning the multiplicative structure, $\Omega_*^{Spin}/Torsion$$\Omega_*^{Spin}/Torsion$ is the subring of an integral polynomial ring on classes $x_{4i}$$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}$$\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

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

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

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

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

$\Omega_3^{Spin}=0$$\Omega_3^{Spin}=0$.

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

$\Omega_5^{Spin}=\Omega_6^{Spin}=\Omega_7^{Spin}=0$$\Omega_5^{Spin}=\Omega_6^{Spin}=\Omega_7^{Spin}=0$.

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

4.2 Generators in all dimensions as given by the classification

By the theorem of Anderson, Brown and Peterson there exist manifolds $M_J$$M_J$ of dimension $4n(J)$$4n(J)$ if $n(J)$$n(J)$ is even, $N_J$$N_J$ of dimension $4n(J)-2$$4n(J)-2$ if $n(J)$$n(J)$ is odd, and $Z_i$$Z_i$ of dimension $|z_i|$$|z_i|$, such that the characteristic numbers $\pi^J(M_J)$$\pi^J(M_J)$, $\pi^J(N_J)$$\pi^J(N_J)$ and $z_i(Z_i)$$z_i(Z_i)$ are odd. For $n(J)$$n(J)$ odd, let $W_J$$W_J$ be a spin nullbordism of $N_J\times \alpha_1$$N_J\times \alpha_1$, and let $M_J=W_J\cup W_J$$M_J=W_J\cup W_J$ (using an orientation-reversing automorphism of $\alpha$$\alpha$). By [Stong1966] there exist manifolds $P_J$$P_J$ such that $4 P_J = M_J\times \tau_4$$4 P_J = M_J\times \tau_4$. If all $j_i$$j_i$ are even, one can choose $M_J$$M_J$ to be a product of quaternionic projective spaces.

Then a basis for $\Omega_*^{Spin}\otimes \Qq$$\Omega_*^{Spin}\otimes \Qq$ is given by

• $M_J\times \omega_8^k$$M_J\times \omega_8^k$ for $k\ge 0$$k\ge 0$
• $M_J\times \tau_4 \times \omega_8^k$$M_J\times \tau_4 \times \omega_8^k$ for $k\ge 0$$k\ge 0$.

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

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

5 Further topics

5.1 Rohlin's theorem

Above we stated that the 4-dimensional spin bordism group $\Omega_4^{Spin} \cong \Z$$\Omega_4^{Spin} \cong \Z$ is generated by the Kummer surface $K3$$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}$$\Omega_2^{Spin}$ give above).

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

5.2 Relationship with other bordism groups

• Framed bordism: the image of $\Omega_*^{fr} \to \Omega_*^{Spin}$$\Omega_*^{fr} \to \Omega_*^{Spin}$ is 0 unless $* = 8k+1$$* = 8k+1$ or $8k+2$$8k+2$ when it is $\Zz_2$$\Zz_2$ and detected by the $\alpha$$\alpha$-invariant. Explicit generators are $\omega_8^k\times \alpha_1^i$$\omega_8^k\times \alpha_1^i$ with $i\le 2$$i\le 2$ and $k\ge 0$$k\ge 0$.

• Oriented bordism: the kernel of $\Omega_*^{Spin} \to \Omega_*^{SO}$$\Omega_*^{Spin} \to \Omega_*^{SO}$ lies in dimensions $8k + 1$$8k + 1$ and $8k + 2$$8k + 2$. It is a $\Zz_2$$\Zz_2$ vector space with a basis $M_J\times \omega_8^i \times\alpha_1^i$$M_J\times \omega_8^i \times\alpha_1^i$, for $n(J)$$n(J)$ even, and $P_J\times \omega_8^i \times\alpha_1^i$$P_J\times \omega_8^i \times\alpha_1^i$, for $n(J)$$n(J)$ odd, where $k\ge 0$$k\ge 0$ and $i\le 2$$i\le 2$. It is also the ideal generated by the non-trivial class of $\Omega_1^{Spin} = \Zz_2$$\Omega_1^{Spin} = \Zz_2$. The cokernel is a finite $2$$2$-torsion group which is trivial if and only if $\Omega^{SO}_\ast = 0$$\Omega^{SO}_\ast = 0$ or equivalently $\ast=1$$\ast=1$, $2$$2$, $3$$3$, $6$$6$, and $7$$7$. [Milnor1965] computed $\Omega^{SO}_8/\Omega^{Spin}_8\cong \Zz_{2^7}$$\Omega^{SO}_8/\Omega^{Spin}_8\cong \Zz_{2^7}$.

• Unoriented bordism: the image of $\Omega_*^{Spin} \to \mathcal{N}_*$$\Omega_*^{Spin} \to \mathcal{N}_*$ is all bordism classes for which the characteristic numbers divisible by $w_1$$w_1$ and $w_2$$w_2$ are zero. A basis for the image consists of the $M_J$$M_J$, for $n(J)$$n(J)$ even, the $N_J$$N_J$, for $n(J)$$n(J)$ odd and the $Z_i$$Z_i$. The image is trivial for $\ast<8$$\ast<8$. In even dimensions it is additionally trivial for $\ast=14$$\ast=14$ and $16$$16$. In odd dimensions it is trivial for $\ast<29$$\ast<29$ and also for $\ast=31$$\ast=31$ and $35$$35$. Otherwise the image is non trivial.

6 References

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