Milnor Hypersurfaces
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== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | .. | + | The signature of the Milnor hypersurfaces is known: |
+ | |||
+ | {{beginthm|Proposition}} \label{prop:signature-milnor} | ||
+ | |||
+ | $$ | ||
+ | \sigma(H_{ij})=\begin{cases} | ||
+ | 1 & : i \textrm{ even, } j \textrm{ odd} \\ | ||
+ | 0 & : \textrm{otherwise} | ||
+ | \end{cases} | ||
+ | $$ | ||
+ | |||
+ | {{beginproof}} | ||
+ | [[Media:signature-milnor.pdf|Click here - opens a separate pdf file]]. | ||
+ | {{endproof}} | ||
+ | |||
</wikitex> | </wikitex> | ||
Revision as of 23:24, 31 May 2012
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
The Milnor hypersurfaces, denoted , are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.
2 Construction and examples
For fixed natural numbers , is defined as the hypersurface in satisfying the equation , where and are homogeneous coordinates for and respectively.
The role of these manifolds in complex bordism is descbribed on the page Complex bordism.
3 Invariants
The signature of the Milnor hypersurfaces is known:
4 References
\leq i \leq j$, $H_{ij}$ is defined as the hypersurface in $\CP^i \times \CP^j$ satisfying the equation $x_0z_0 + ... + x_iz_i = 0$, where $x_k$ and $z_k$ are homogeneous coordinates for $\CP^i$ and $\CP^j$ respectively. The role of these manifolds in complex bordism is descbribed on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]]. == Invariants ==2 Construction and examples
For fixed natural numbers , is defined as the hypersurface in satisfying the equation , where and are homogeneous coordinates for and respectively.
The role of these manifolds in complex bordism is descbribed on the page Complex bordism.
3 Invariants
The signature of the Milnor hypersurfaces is known: