Spin bordism
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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
). By [Stong1966] there exist manifolds
such that
. 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_*(pt).](/images/math/0/3/b/03bbd30ed92f4a4a25beab5feea39676.png)
A feature of Spin manifolds is that they possess Dirac operators,
. The (Clifford-linear) Dirac operator can be considered as a representative of the fundamental class
,
where
is the dimension of
, see [Atiyah1970] and [Higson&Roe2000].
The characteristic numbers above can then be defined as the indices of the (Clifford-linear) Dirac operators obtained by twisting 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 [\operatorname{ind} \partial\!\!\!/_M]=\alpha(M)=\pi^0(M).](/images/math/5/b/e/5befb82949e3197c345ca4ce92f695f5.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
5.1 Relationship with other bordism groups
- Framed bordism the image of
is 0 except in unless
or
when it is
and detected by the
-invariant. Explicit generators are
with
and
.
- Oriented bordism: the kernel of
lies in dimensions
and
.It is a
vector space with a basis
,
even, and
,
odd where
and
. It is also the ideal generated by the non-trivial class of
. The cokernel is a finite
-torsion group which is trivial if and only of
or equivalently
,
,
,
, and
. [Milnor1965] computed
.
- Unoriented bordism: the image of
is all bordism classes for which the characteristic numbers divisible by
and
are zero. A basis for the image consists of the
,
even, the
,
odd and the
. The first
occurs in dimension
. The image is trivial for
. In even dimensions it is additionally trivial for
and
. In odd dimensions it is trivial for
and also for
and
. Otherwise the image is non trivial. Indeed, there are
in all dimensions
.
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
- [Atiyah1970] M. F. Atiyah, Global theory of elliptic operators, (1970), 21–30. MR0266247 (42 #1154) Zbl 0193.43601
- [Higson&Roe2000] N. Higson and J. Roe, Analytic
-homology, Oxford University Press, Oxford, 2000. MR1817560 (2002c:58036) Zbl 1146.19004
- [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
- [Milnor1965] J. W. Milnor, Remarks concerning spin manifolds, in Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse, (1965) 55–62. MR0180978 (31 #5208) Zbl 0132.19602
- [Stong1966] R. E. Stong, Relations among characteristic numbers. II, Topology 5 (1966), 133–148. MR0192516 (33 #741) Zbl 0142.40902
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010