Milnor Hypersurfaces

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1 Introduction

The Milnor hypersurfaces, denoted $H_{ij}$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; The Milnor hypersurfaces, denoted $H_{ij}$, are a family of smooth manifolds that generate (with redundancy) the [[Complex bordism|complex bordism]] ring. </wikitex> == Construction and examples == <wikitex>; For fixed natural numbers $i \geq j \geq 0$, $H_{ij}$ is defined as the hypersurface in $\CP^i \times \CP^j$ satisfying the equation $x_0z_0 + ... + x_jz_j = 0$, where $x_k$ and $z_k$ are homogeneous coordinates for $\CP^i$ and $\CP^j$ respectively. </wikitex> == References == {{#RefList:}} [[Complex bordism#Milnor hypersurfaces]] -- the treatment here is excellent, but it's good to have a separate page for reference purposes. [[Category:Manifolds]]H_{ij}$, are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.

2 Construction and examples

For fixed natural numbers $i \geq j \geq 0$$i \geq j \geq 0$, $H_{ij}$$H_{ij}$ is defined as the hypersurface in $\CP^i \times \CP^j$$\CP^i \times \CP^j$ satisfying the equation $x_0z_0 + ... + x_jz_j = 0$$x_0z_0 + ... + x_jz_j = 0$, where $x_k$$x_k$ and $z_k$$z_k$ are homogeneous coordinates for $\CP^i$$\CP^i$ and $\CP^j$$\CP^j$ respectively.

3 References

Complex bordism#Milnor hypersurfaces -- the treatment here is excellent, but it's good to have a separate page for reference purposes.