Unoriented bordism
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\mathcal N_* := \sum _i\mathcal N_i | \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 | + | 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. |
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Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $\mathcal N_*$. | Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $\mathcal N_*$. | ||
− | {{beginthm|Theorem (Dold) {{cite|Dold1956}} | + | {{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,... | x_2,x_4,x_5,x_6,x_8,... |
Revision as of 16:53, 8 January 2010
The user responsible for this page is Matthias Kreck. No other user may edit this page at present. |
Contents |
1 Introduction
We denote the non-oriented bordism groups by . 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 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
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
are polynomial generators of olver :
3 Invariants
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4 Classification/Characterization (if available)
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5 Further discussion
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6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , 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. |