Unoriented bordism
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Contents |
1 Introduction
We denote the unoriented bordism groups by . The sum of these groups
![\displaystyle \mathcal N_* := \sum _i\mathcal N_i](/images/math/0/0/3/0034cbe1717e4b2495768cda2e75c4b2.png)
forms a ring under cartesian products of manifolds. Thom [Thom1954] has shown that this ring is a polynomial ring over in variables
for
and he has shown that for
even one can take
for
. Dold [Dold1956] has constructed manifolds for
with
odd.
2 Construction and examples
Dold constructs certain bundles over with fibre
denoted by
![\displaystyle P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,](/images/math/c/2/3/c23bfe20949022b58e4dbbdd036548fc.png)
![\tau](/images/math/2/4/f/24f649f2eaad83d8a6a97f8e49fc6fac.png)
![(x,[y])](/images/math/5/f/4/5f4498f04b89f91ead4518908defc4a5.png)
![(-x, [\bar y])](/images/math/d/b/a/dba628c4f8c13c856cc15d4ca98d20dd.png)
![\bar y = (\bar y_0,...,\bar y_n)](/images/math/b/e/b/bebea2bf185550fa4707926e5742bdb1.png)
![y =(y_0,...y_n)](/images/math/c/1/4/c14239669bb6396066e9a7a3aacb9466.png)
Using the results by Thom [Thom1954] Dold shows that these manifolds give ring generators of .
Theorem (Dold) [Dold1956] 2.1. For even set
and for
set
. Then for
![\displaystyle x_2,x_4,x_5,x_6,x_8,...](/images/math/6/d/e/6de6b7f616109ed997e9dd27b8dd4842.png)
are polynomial generators of olver
:
![\displaystyle \mathcal N_* \cong \mathbb Z/2[x_2,x_4,x_5,x_6,x_8...].](/images/math/b/2/8/b2808ca7f9cef2cc7632e4d8c8d31bd2.png)
3 Invariants
To prove the Theorem Dold has to compute the characteristic numbers which according to Thom's theorem determine the bordism class. As a first step Dold computes the cohomology ring with -coeffcients. The fibre bundle
has a section
and we consider the cohomology classes (always with
-coefficients)
![\displaystyle c:= p^*(x) \in H^1(P(m,n)),](/images/math/6/2/b/62ba37a98c8a60ab20164bea4d1cba4c.png)
where is a generator of
, and
![\displaystyle d \in H^2(P(m,n)),](/images/math/0/2/b/02b0a35923edc8d5f247635fadd15822.png)
which is characterized by the property that the restriction to a fibre is non-trivial and .
Theorem [Dold1956] 3.1. The classes and
generate
with only the relations
![\displaystyle c^{m+1} =0](/images/math/7/2/4/724552bc4500d93dafbf31748e744351.png)
and
![\displaystyle d^{n+1} =0.](/images/math/1/c/4/1c4ce2ab47fc9a001369be7c1f78f652.png)
The Steenrod squares act by
![\displaystyle Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,](/images/math/2/9/e/29eb4f3dc137e3d530f3ab316ce1dc5c.png)
and all other Squares act trivially on
and
. On the decomposable classes the action is given by the Cartan formula.
The total Stiefel-Whitney class of the tangent bundle is
![\displaystyle w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}.](/images/math/3/b/7/3b713825bda2917b4c4c0f750c9b496a.png)
4 Classification
To give explicit polynomial generators is useful information, if one wants to prove a formula like for example that , where
is the
-th Stiefel-Whitney class of an
-dimensional manifold and
is the Euler characteristic, which one has to check on generators. But it does not help to classify manifolds up to bordism. There is an explicit answer to this question by Thom:
Theorem [Thom1954] 4.1. Two closed -manifolds
and
are bordant if and only if all Stiefel-Whitney numbers agree:
![\displaystyle <w_{i_1}\cup....\cup w_{i_k}(M), [M]> = <w_{i_1}\cup....\cup w_{i_k}(N), [N]>](/images/math/d/1/0/d10c5e54476446995c8b94e52914f356.png)
for all partitions .
5 Further discussion
For odd
the manifolds
are orientable and thus after choosing an orientation give an element in the oriented bordism group
. Since
admits an obvious orientation reversing diffeomorphism, these elements are
-torsion. Thus we obtain a subring in
isomorphic to
. For more information about
see the page on oriented bordism.
6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra
, Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
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