Unoriented bordism

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1 Introduction

We denote the non-oriented bordism groups by $\mathcal N_i$${{Authors|Matthias Kreck}} == Introduction == <wikitex>; We denote the non-oriented bordism groups by $\mathcal N_i$. The sum of these groups $$ \mathcal N_* := \sum _i\mathcal N_i $$ are a ring under cartesian products of the manifolds. Thom {{cite|Thom1954}} has shown that this ring is a polynomial ring over $\mathbb Z/2$ in variables $x_i$ for $i \ne 2^k -1$ and he has shown that for $i$ even one can take $\mathbb {RP}^i$ for $x_i$. Dold {{cite|Dold1956}} has constructed manifolds for $x_i$ with $i $ odd. </wikitex> == Construction and examples == <wikitex>; Dold constructs certain bundles over $\mathbb {RP}^m$ with fibre $\mathbb {CP}^n$ denoted by $$ P(m,n):= (S^m \times \mathbb {CP}^m)/\tau, $$ where $\tau$ is the involution mapping $(x,[y]) $ to $(-x, [\bar y])$ and $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$. These manifolds are now cold Dold manifolds. Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $\mathcal N_*$. {{beginthm|Theorem (Dold) {{cite|Dold1956}}}} For $i$ even set $x_i:= [P(i,0) ]= [\mathbb {RP}^i]$ and for $i = 2^r(2s+1)-1$ set $x_i:=[ P(2^r-1,s2^r)]$. Then for $i \ne 2^k-1$ $$ x_2,x_4,x_5,x_6,x_8,... $$ are polynomial generators of $\mathcal N_*$ olver $\mathbb Z/2$: $$ \mathcal N_* \cong \mathbb Z/2[x_2,x_4,x_5,x_6,x_8...].$$ {{endthm}} </wikitex> == Invariants == <wikitex>; To prove the Theorem Dold has to compute the characteristic numbers which according to Thom's theorem determine the bordism class. As a first step Dold computes the cohomology ring with $\mathbb Z/2$-coeffcients. The fibre bundle $p:P(m,n) \to \mathbb {RP}^m$ has a section $s([x]) := [(x,[1,...,1])]$ and we consider the cohomology classes (always with $\mathbb Z/2$-coefficients) $$ c:= p^*(x) \in H^1(P(m,n)), $$ where $x$ is a generator of $H^1(\mathbb {RP}^m)$, and $$ d \in H^2(P(m,n)), $$ which is characterized by the property that the restriction to a fibre is non-trivial and $s^*(d)=0$. {{beginthm|Theorem {{cite|Dold1956}}}} The classes $c \in H^1(P(m,n))$ and $d\in H^2(P(m,n))$ generate $H^*(P(m,n);\mathbb Z/2)$ with only relation $$ c^{m+1} =0 $$ and $$ d^{n+1} =0. $$ The Steenrod squares act by $$ Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2, $$ and all other Squares $Sq^i$ act trivially on $c$ and $d$. On the decomposable classes the action is given by the Cartan formula. The total Stiefel-Whitney class of the tangent bundle is $$ w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}. $$ {{endthm}} </wikitex> == Classification/Characterization (if available) == <wikitex>; YOUR TEXT HERE ... </wikitex> == Further discussion == <wikitex>; For $i= 2^r(2s+1)-1$ odd $i \ne 2^k-1$ the manifolds $P_i:= P(2^r-1),s2^r)$ are orientable and thus after choosing an orientation give an element in the oriented bordism group $\Omega_i$. Since $P_i$ admits an obvious orientation reversing diffeomorphism, these elements are $-torsion. Thus we obtain a subring in $\Omega_*$ isomorphic to $\mathbb Z/2[x_5, x_9, x_{11},...]$. For more information about $\Omega _*$ see the page on oriented bordism.</wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\mathcal N_i$. The sum of these groups

$\displaystyle \mathcal N_* := \sum _i\mathcal N_i$ $$\displaystyle \mathcal N_* := \sum _i\mathcal N_i$$

are a ring under cartesian products of the manifolds. Thom [Thom1954] has shown that this ring is a polynomial ring over $\mathbb Z/2$$\mathbb Z/2$ in variables $x_i$$x_i$ for $i \ne 2^k -1$$i \ne 2^k -1$ and he has shown that for $i$$i$ even one can take $\mathbb {RP}^i$$\mathbb {RP}^i$ for $x_i$$x_i$. Dold [Dold1956] has constructed manifolds for $x_i$$x_i$ with $i$$i$ odd.

2 Construction and examples

Dold constructs certain bundles over $\mathbb {RP}^m$$\mathbb {RP}^m$ with fibre $\mathbb {CP}^n$$\mathbb {CP}^n$ denoted by

$\displaystyle P(m,n):= (S^m \times \mathbb {CP}^m)/\tau,$ $$\displaystyle P(m,n):= (S^m \times \mathbb {CP}^m)/\tau,$$

where $\tau$$\tau$ is the involution mapping $(x,[y])$$(x,[y])$ to $(-x, [\bar y])$$(-x, [\bar y])$ and $\bar y = (\bar y_0,...,\bar y_n)$$\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$$y =(y_0,...y_n)$. These manifolds are now cold Dold manifolds.

Using the results by Thom [Thom1954] Dold shows that these manifolds give ring generators of $\mathcal N_*$$\mathcal N_*$.

Theorem (Dold) [Dold1956] 2.1. For $i$$i$ even set $x_i:= [P(i,0) ]= [\mathbb {RP}^i]$$x_i:= [P(i,0) ]= [\mathbb {RP}^i]$ and for $i = 2^r(2s+1)-1$$i = 2^r(2s+1)-1$ set $x_i:=[ P(2^r-1,s2^r)]$$x_i:=[ P(2^r-1,s2^r)]$. Then for $i \ne 2^k-1$$i \ne 2^k-1$

$\displaystyle x_2,x_4,x_5,x_6,x_8,...$ $$\displaystyle x_2,x_4,x_5,x_6,x_8,...$$

are polynomial generators of $\mathcal N_*$$\mathcal N_*$ olver $\mathbb Z/2$$\mathbb Z/2$:

$\displaystyle \mathcal N_* \cong \mathbb Z/2[x_2,x_4,x_5,x_6,x_8...].$ $$\displaystyle \mathcal N_* \cong \mathbb Z/2[x_2,x_4,x_5,x_6,x_8...].$$

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 $\mathbb Z/2$$\mathbb Z/2$-coeffcients. The fibre bundle $p:P(m,n) \to \mathbb {RP}^m$$p:P(m,n) \to \mathbb {RP}^m$ has a section $s([x]) := [(x,[1,...,1])]$$s([x]) := [(x,[1,...,1])]$ and we consider the cohomology classes (always with $\mathbb Z/2$$\mathbb Z/2$-coefficients)

$\displaystyle c:= p^*(x) \in H^1(P(m,n)),$ $$\displaystyle c:= p^*(x) \in H^1(P(m,n)),$$

where $x$$x$ is a generator of $H^1(\mathbb {RP}^m)$$H^1(\mathbb {RP}^m)$, and

$\displaystyle d \in H^2(P(m,n)),$ $$\displaystyle d \in H^2(P(m,n)),$$

which is characterized by the property that the restriction to a fibre is non-trivial and $s^*(d)=0$$s^*(d)=0$.

Theorem [Dold1956] 3.1. The classes $c \in H^1(P(m,n))$$c \in H^1(P(m,n))$ and $d\in H^2(P(m,n))$$d\in H^2(P(m,n))$ generate $H^*(P(m,n);\mathbb Z/2)$$H^*(P(m,n);\mathbb Z/2)$ with only relation

$\displaystyle c^{m+1} =0$ $$\displaystyle c^{m+1} =0$$

and

$\displaystyle d^{n+1} =0.$ $$\displaystyle d^{n+1} =0.$$

The Steenrod squares act by

$\displaystyle Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,$ $$\displaystyle Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,$$

and all other Squares $Sq^i$$Sq^i$ act trivially on $c$$c$ and $d$$d$. On the decomposable classes the action is given by the Cartan formula.

The total Stiefel-Whitney class of the tangent bundle is

$\displaystyle w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}.$ $$\displaystyle w(P(m,n)) = (1+c)^{m+1}(1+d)^{n+1}.$$

4 Classification/Characterization (if available)

YOUR TEXT HERE ...

5 Further discussion

For $i= 2^r(2s+1)-1$$i= 2^r(2s+1)-1$ odd $i \ne 2^k-1$$i \ne 2^k-1$ the manifolds $P_i:= P(2^r-1),s2^r)$$P_i:= P(2^r-1),s2^r)$ are orientable and thus after choosing an orientation give an element in the oriented bordism group $\Omega_i$$\Omega_i$. Since $P_i$$P_i$ admits an obvious orientation reversing diffeomorphism, these elements are $2$$2$-torsion. Thus we obtain a subring in $\Omega_*$$\Omega_*$ isomorphic to $\mathbb Z/2[x_5, x_9, x_{11},...]$$\mathbb Z/2[x_5, x_9, x_{11},...]$. For more information about $\Omega _*$$\Omega _*$ see the page on oriented bordism.

6 References

• [Dold1956] A. Dold, Erzeugende der Thomschen Algebra $\mathcal{N}$$\mathcal{N}$, 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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