Unoriented bordism
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Contents |
1 Introduction
We denote the non-oriented bordism groups by . The sum of these groups
![\displaystyle \mathcal N_* := \sum _i\mathcal N_i](/images/math/0/0/3/0034cbe1717e4b2495768cda2e75c4b2.png)
are a ring under cartesian products of the 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 relation
![\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/Characterization (if available)
YOUR TEXT HERE ...
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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