Spin bordism
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{{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}} | {{beginthm|Theorem {{cite|Anderson&Brown&Peterson1966}}} | ||
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There is a 2-local homotopy equivalence | There is a 2-local homotopy equivalence | ||
$$MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle | $$MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle |
Revision as of 22:29, 26 January 2010
Contents |
1 Introduction
The spin bordism groups of manifolds with spin structures are the homotopy groups of the Thom spectrum .
2 Calculation
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 classes
%This should probably get its own page.%
The KO-Pontryagin classes are defined by setting for for complex line bundles L and then requiring naturality and where . Here and are oriented bundles. In fact, these properties determine because the group injects into under the complexification of the map which is induced by the restriction to the maximal torus (compare [Anderson&Brown&Peterson1966]). For we set . Such a class gives a map .
{{beginthm|Theorem [Anderson&Brown&Peterson1966]}
There is a 2-local homotopy equivalence
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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
This page has not been refereed. The information given here might be incomplete or provisional. |