Milnor Hypersurfaces

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1 Introduction

The Milnor hypersurfaces, denoted $H_{ij}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}H_{ij}$, are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.

2 Construction and examples

For fixed natural numbers $0 \leq i \leq j$$0 \leq i \leq j$, $H_{ij}$$H_{ij}$ is defined as the hypersurface in $\CP^i \times \CP^j$$\CP^i \times \CP^j$ satisfying the equation $x_0z_0 + ... + x_iz_i = 0$$x_0z_0 + ... + x_iz_i = 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. This equation defines a generic hyperplane intersecting the image of the Segre embedding $\CP^i \times \CP^j \to \CP^{(i+1)(j+1)-1}$$\CP^i \times \CP^j \to \CP^{(i+1)(j+1)-1}$ transversely.

The role of these manifolds in complex bordism is described on the page Complex bordism.

3 Invariants

The signature of the Milnor hypersurfaces is known:

Proposition 3.1.

$\displaystyle \sigma(H_{ij})=\begin{cases} 1 & i \textrm{ even, } j \textrm{ odd} \\ 0 & \textrm{otherwise} \end{cases}$ $$\displaystyle \sigma(H_{ij})=\begin{cases} 1 & i \textrm{ even, } j \textrm{ odd} \\ 0 & \textrm{otherwise} \end{cases}$$

Proof. Click here - opens a separate pdf file.

$\square$$\square$