Spin bordism
Line 3: | Line 3: | ||
The spin bordism groups $\Omega_n^{Spin}$ of manifolds with spin structures are the homotopy groups of the Thom spectrum $MSpin$. | The spin bordism groups $\Omega_n^{Spin}$ of manifolds with spin structures are the homotopy groups of the Thom spectrum $MSpin$. | ||
− | |||
Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson | Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson | ||
− | {{cite|Anderson&Brown& | + | {{cite|Anderson&Brown&Peterson1967}} 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. |
− | 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. | + | |
</wikitex> | </wikitex> | ||
Line 32: | Line 30: | ||
for manifolds of smaller dimension. | for manifolds of smaller dimension. | ||
− | {{beginthm|Theorem {{cite|Anderson&Brown& | + | {{beginthm|Theorem {{cite|Anderson&Brown&Peterson1967}}}} |
There are classes $x_i\in H^*(MSpin;\Zz_2)$ such that there is a 2-local homotopy equivalence | There are classes $x_i\in H^*(MSpin;\Zz_2)$ such that there is a 2-local homotopy equivalence | ||
$$(\pi^J,x_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle | $$(\pi^J,x_i): MSpin \to\bigvee_{n(J)even, \\ 1\not \in J}ko \langle 4n(J)\rangle | ||
Line 40: | Line 38: | ||
\bigvee_{i}\Sigma^{|x_i|}H\Zz_2$$ | \bigvee_{i}\Sigma^{|x_i|}H\Zz_2$$ | ||
{{endthm}} | {{endthm}} | ||
+ | |||
+ | Stong proves that $\Omega_*^{Spin}/Torsion$ 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 {{cite|Stong1968}}. | ||
+ | Anderson, Brown and Peterson determine the structure of $\Omega_*^{Spin}$ modulo | ||
+ | torsion mapping monomorphically into unoriented cobordism. | ||
== Generators == | == Generators == |
Revision as of 21:13, 27 January 2010
Contents |
1 Introduction
The spin bordism groups of manifolds with spin structures are the homotopy groups of the Thom spectrum .
Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson [Anderson&Brown&Peterson1967] 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.
2 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers. (Since a spin structure induces a KO-orientation, we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbers in .) These characteristic numbers can be defined as the indices of Clifford-linear Dirac operators twisted with the corresponding vector bundles. See [Lawson&Michelsohn1989].
3 Classification
is an equivalence after inverting 2. Thus there is no odd torsion in the spin cobordism groups.
For where all , 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. Hence the corresponding characteristic numbers vanish for manifolds of smaller dimension.
Theorem [Anderson&Brown&Peterson1967] 3.1. There are classes such that there is a 2-local homotopy equivalence
Stong proves that 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 [Stong1968]. Anderson, Brown and Peterson determine the structure of modulo torsion mapping monomorphically into unoriented cobordism.
Generators
, generated by a point.
, generated by , the circle with the "antiperiodic" spin structure.
, generated by .
.
, generated by , the Kummer surface.
.
, generated by quaternionic projective space, and a generator which equals 1/4 of the square of the Kummer surface.
By the above theorem of Anderson, Brown and Peterson there exist manifolds
- of dimension if is even,
- of dimension if is odd,
such that the characteristic numbers , and are odd. For odd, let be a spin nullbordism of , and let (using an orientation-reversing automorphism of ).
Then a basis for is given by
- .
A basis for is given by
- with and even,
- with even,
- with odd,
- with and odd
- ,
- .
4 Further topics
Relation to framed bordism: The image of framed cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$.
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.
4.1 References
- [Anderson&Brown&Peterson1967] D. W. Anderson, E. H. Brown and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR0219077 (36 #2160) Zbl 0156.21605
- [Lawson&Michelsohn1989] H. B. Lawson and M. Michelsohn, Spin geometry, Princeton University Press, Princeton, NJ, 1989. MR1031992 (91g:53001) Zbl 0801.58017
- [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. |