Unoriented bordism

The user responsible for this page is Matthias Kreck. No other user may edit this page at present.

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 $$ are a ring under cartesian products of the manifolds. Thom \cite{Thom} has shown that this ring is a polynomial ring over $\mathbb Z/2$ in variables $x_i$ for $i \ne 2^k -1$ and he has shown that for $i$ even one can take $\mathbb {RP}^i$ for $x_i$. Dold {{cite|Dold1956}} has constructed manifolds for $x_i$ with $i $ odd. </wikitex> == Construction and examples == <wikitex>; Dold constructs certain bundles over $\mathbb {RP}^m$ with fibre $\mathbb {CP}^n$ denoted by $$ 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>; YOUR TEXT HERE ... </wikitex> == Classification/Characterization (if available) == <wikitex>; YOUR TEXT HERE ... </wikitex> == Further discussion == <wikitex>; YOUR TEXT HERE ... </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 [Thom] 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

YOUR TEXT HERE ...

4 Classification/Characterization (if available)

YOUR TEXT HERE ...

5 Further discussion

YOUR TEXT HERE ...

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
• [Thom] Template:Thom
• [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502

This page has not been refereed. The information given here might be incomplete or provisional.