Unoriented bordism

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	== Introduction ==		== Introduction ==
	<wikitex>;		<wikitex>;
−	YOUR TEXT HERE ...	+	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 {Dold} has constructed manifolds for $x_i$ with $i $ odd.
		+
	</wikitex>		</wikitex>
	== Construction and examples ==		== Construction and examples ==
	<wikitex>;		<wikitex>;
−	YOUR TEXT HERE ...	+	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{Thom} Dold shows that these manifolds give ring generators of $\mathcal N_*$.
		+
		+	{{beginthm|Theorem (Dold) \cite{Dold}:}} 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>		</wikitex>

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Revision as of 12:36, 8 January 2010

Contents 1 Introduction 2 Construction and examples 3 Invariants 4 Classification/Characterization (if available) 5 Further discussion 6 References

1 Introduction

We denote the non-oriented bordism groups by $== Introduction == <wikitex>; YOUR TEXT HERE ... </wikitex> == Construction and examples == <wikitex>; YOUR TEXT HERE ... </wikitex> == Invariants == <wikitex>; YOUR TEXT HERE ... </wikitex> == Classification/Characterization (if available) == <wikitex>; YOUR TEXT HERE ... </wikitex> == Further discussion == <wikitex>; YOUR TEXT HERE ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\mathcal N_i$. The sum of these groups

are a ring under cartesian products of the manifolds. Thom [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 {Dold} has constructed manifolds for $x_i$ with $i$ odd.

2 Construction and examples

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

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 [Thom] Dold shows that these manifolds give ring generators of $\mathcal N_*$.

3 Invariants

YOUR TEXT HERE ...

4 Classification/Characterization (if available)

YOUR TEXT HERE ...

5 Further discussion

YOUR TEXT HERE ...

6 References

• [Dold] Template:Dold
• [Thom] Template:Thom

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