Spin bordism
Contents |
1 Introduction
By the Pontrjagin-Thom isomorphism the spin bordism groups of closed manifolds with spin structures are isomorphic to the homotopy groups of the Thom spectrum
.
Preliminary results were by Novikov. The main calculation was achieved in
[Anderson&Brown&Peterson1966] and [Anderson&Brown&Peterson1967] where it is shown that all torsion is of exponent 2, being of two types: that arising by products with a framed and that which maps monomorphically into unoriented cobordism.
2 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.
For a multi-index where all
, we set
and
.
By the theorem of Anderson, Brown and Peterson below 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
). If all
are even,
one can choose
to be a product of quaternionic projective spaces.
Then a basis for is given by
-
-
.
A basis for is given by
-
with
and
even,
-
with
even,
-
with
odd,
-
with
and
odd
-
,
-
.
3 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers.
![[M]](/images/math/f/a/0/fa08c3d5d2f54260952acc8a646b5025.png)
![\pi^j](/images/math/4/4/4/444981b53dfb30d4c9b37c087cc80365.png)
![\displaystyle \pi^J(M)=\langle \pi^J(TM), [M] \rangle \in KO_n(pt).](/images/math/e/f/1/ef11fdaca6fe169cbb99a2a7b7bc5cb1.png)
A feature of Spin manifolds is that they possess Dirac operators,
.
The characteristic numbers above can be defined as the indices of (Clifford-linear) Dirac operators twisted with the corresponding vector bundles. The easiest case is the non-twisted one:
is the trivial bundle, and taking the index of the Dirac operator
defines an element of
when
is n-dimensional. This gives rises to a ring homomorphism often called the
-invariant:
![\displaystyle \alpha : \Omega_*^{Spin} \longrightarrow KO_*, ~~~ [M] \longmapsto [\partial\!\!\!/_M]=\alpha(M)=\pi^0(M).](/images/math/3/e/b/3eb7162ed9d4897bfe4eb59c9296e6b1.png)
See [Lawson&Michelsohn1989].
4 Classification
After inverting 2 the map of Thom spectra becomes a homotopy equivalence. Thus there is no odd torsion in the spin cobordism groups, and all
summands are in degrees divisible by 4.
Using the Thom isomorphism we get for each multi-index
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] 4.1.
There are classes such that there is a 2-local homotopy equivalence
![\displaystyle (\pi^J,z_i): 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^{|z_i|}H\Zz_2.](/images/math/9/1/d/91d11a635e36d336701efcb8c2c2027c.png)
From this one can compute the additive structure completely.
Concerning the multiplicative structure, 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
the ideal consisting of torsion mapping monomorphically into unoriented cobordism.
According to [Laures2003], the multiplicative structure of this ideal is still not completely known.
5 Further topics
Relation to framed bordism: The image of framed bordism is 0 except in dimensions where it is
.
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 bordism is all classes for which the characteristic numbers divisible by and
are zero.
6 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
- [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
- [Laures2003] G. Laures, An
splitting of spin bordism, Amer. J. Math. 125 (2003), no.5, 977–1027. MR2004426 (2004g:55007) Zbl 1058.55001
- [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
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