Spin bordism

Revision as of 22:44, 26 January 2010

1 Introduction

The spin bordism groups $== Introduction == <wikitex>; The spin bordism groups $\Omega_n^{Spin}$ of manifolds with spin structures are the homotopy groups of the Thom spectrum $MSpin$. </wikitex> == Calculation == <wikitex>; Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson {{cite|Anderson&Brown&Peterson1966}} who showed that all torsion is of order 2, being of two types: that arising by products with a framed $S^1$ and that which maps monomorphically into unoriented cobordism. $\Omega_*^{Spin}/Torsion$ 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. Characteristic numbers: Cobordism is determined by $\Zz_2$ cohomology and $KO$-theory characteristic numbers. Relation to framed bordism: The image of framed cobordism is 0 except in dimensions k+1,8k+2$ where it is $\Zz_2$. Relation to oriented bordism: $MSpin\to MSO$ is an equivalence after inverting 2. The kernel of the map from spin to oriented bordism is in dimensions k + 1$ and k + 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 cobordism is all classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero. </wikitex> == KO-Characteristic classes == <wikitex>; %This should probably get its own page.% The KO-Pontryagin classes $\pi^j$ are defined by setting $\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$ for complex line bundles L and then requiring naturality and $\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$ where $\pi_s =\sum_j \pi^j s^j$ . Here $\xi$ and $\eta$ are oriented bundles. In fact, these properties determine $\pi^j$ because the group $KO(BSO(m))$ injects into $K(BT^{[m/2]})$ under the complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$ (compare {{cite|Anderson&Brown&Peterson1966}}). For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$. Such a class gives a map $MSpin\to ko \langle m\rangle$. {{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}} There is a 2-local homotopy equivalence $$MSpin \to\bigvee_{n(J)even, \ 1\not \in J}ko \langle 4n(J)\rangle \vee \bigvee_{n(J)odd, \ 1\not \in J}ko \langle 4n(J)+2 \rangle \vee \bigvee_{i}\Sigma^{|x_i|}H\Zz_2$$ {{endthm}} </wikitex> == Low dimensions == <wikitex>; $\Omega_0^{Spin}=\Zz$, generated by a point. $\Omega_1^{Spin}=\Zz_2$, generated by the circle with the "antiperiodic" spin structure. $\Omega_2^{Spin}=\Zz_2$, generated by the square of the generator of the first bordism group. $\Omega_3^{Spin}=0$. $\Omega_4^{Spin}=\Zz$, generated by 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 1/4 of the square of the Kummer surface. </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\Omega_n^{Spin}$ of manifolds with spin structures are the homotopy groups of the Thom spectrum $MSpin$.

2 Calculation

[Stong1968] Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson [Anderson&Brown&Peterson1966] who showed that all torsion is of order 2, being of two types: that arising by products with a framed $S^1$ and that which maps monomorphically into unoriented cobordism. $\Omega_*^{Spin}/Torsion$ 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.

Characteristic numbers: Cobordism is determined by $\Zz_2$ cohomology and $KO$-theory characteristic numbers.

Relation to framed bordism: The image of framed cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$.

Relation to oriented bordism: $MSpin\to MSO$ is an equivalence after inverting 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 cobordism is all classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero.

3 KO-Characteristic classes

%This should probably get its own page.%

The KO-Pontryagin classes $\pi^j$ are defined by setting $\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$ for complex line bundles L and then requiring naturality and $\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$ where $\pi_s =\sum_j \pi^j s^j$ . Here $\xi$ and $\eta$ are oriented bundles. In fact, these properties determine $\pi^j$ because the group $KO(BSO(m))$ injects into $K(BT^{[m/2]})$ under the complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$ (compare [Anderson&Brown&Peterson1966]). For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$. Such a class gives a map $MSpin\to ko \langle m\rangle$.

Theorem [Anderson&Brown&Peterson1966] 3.1. There is a 2-local homotopy equivalence

$$\displaystyle MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle \vee \bigvee_{n(J)odd, \\ 1\not \in J}ko \langle 4n(J)+2 \rangle \vee \bigvee_{i}\Sigma^{|x_i|}H\Zz_2$$

4 Low dimensions

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

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

$\Omega_2^{Spin}=\Zz_2$, generated by the square of the generator of the first bordism group.

$\Omega_3^{Spin}=0$.

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

5 References

• [Anderson&Brown&Peterson1966] D. W. Anderson, E. H. Brown and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR0190939 (32 #8349) Zbl 0156.21605
• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010