Spin bordism
(→KO-Characteristic numbers) |
|||
Line 33: | Line 33: | ||
For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$. | For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$. | ||
− | Using the Thom isomorphism $KO(BSpin)\cong KO(MSpin)$ we get a map $\pi^J:MSpin\to KO$. | + | Using the Thom isomorphism $KO(BSpin)\cong KO(MSpin)$ we get 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. | ||
{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}} | {{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}}} | ||
Line 39: | Line 41: | ||
$$(\pi^J,x_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle | $$(\pi^J,x_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle | ||
\vee | \vee | ||
− | \bigvee_{n(J)odd, \\ 1\not \in J}ko \langle 4n(J) | + | \bigvee_{n(J)odd, \\ 1\not \in J}ko \langle 4n(J)-2 \rangle |
\vee | \vee | ||
\bigvee_{i}\Sigma^{|x_i|}H\Zz_2$$ | \bigvee_{i}\Sigma^{|x_i|}H\Zz_2$$ | ||
Line 45: | Line 47: | ||
</wikitex> | </wikitex> | ||
− | |||
== Low dimensions == | == Low dimensions == |
Revision as of 18:27, 27 January 2010
Contents |
1 Introduction
The spin bordism groups of manifolds with spin structures are the homotopy groups of the Thom spectrum .
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 and that which maps monomorphically into unoriented cobordism. is the subring of an integral polynomial ring on classes (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 cohomology and -theory characteristic numbers.
Relation to framed bordism: The image of framed cobordism is 0 except in dimensions where it is .
Relation to oriented bordism: is an equivalence after inverting 2. The kernel of the map from spin to oriented bordism is in dimensions and 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 and are zero.
3 KO-Characteristic numbers
Since Spin-manifolds have a natural KO-orientation, we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbers.
For we set and . Using the Thom isomorphism we get a map , for which Anderson,Brown and Peterson show that it factorizes through if is even and if is odd.
Theorem [Anderson&Brown&Peterson1966] 3.1. There are classes such that there is a 2-local homotopy equivalence
4 Low dimensions
, generated by a point.
, generated by the circle with the "antiperiodic" spin structure.
, generated by the square of the generator of the first bordism group.
.
, generated by the Kummer surface.
.
, 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
This page has not been refereed. The information given here might be incomplete or provisional. |