Grassmann manifolds

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1 Introduction

Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $K$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}}K$ be the real, complex or quaternion field and $V$$V$ a vector space over $K$$K$. A Grassmann manifolds of $k$$k$-dimensional subspaces is a set $G_k(V)$$G_k(V)$ of $k$$k$-dimensional subspaces. The set $G_k(V)$$G_k(V)$ is a quotient of a subset of $V\times ...\times V$$V\times ...\times V$ consisting of linearly independent $k$$k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$$G_k(V)$ as the quotient topology.

Theorem 1.1 [[[#|]]]. $G_k(V)$$G_k(V)$ is a Hausdorff, compact space. \end{zad}

\begin{zad} Prove that there exist a homeomorphism

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$G_k(\F^n)\simeq G_{n-k}(\F^n)$ \end{zad} \begin{zad} Prove that

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$G_k(\F^n)$ is a connected, compact manifold of dimension $dk(n-k)$$dk(n-k)$. \end{zad} \begin{zad} There is an embedding of the Grassmannian

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$G_k(\F^n)$ in the cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$$\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on

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$G_k(\F^n)$.

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\begin{zad} The Grassmannians

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$G_1(\F^n)$ are the well-known projective spaces, denoted

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$\F P(n)$. Note that

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$G_1(\F^2)=S^d$ and if we identify $S^d$$S^d$ with the one-point compactification of $\F$$\F$ the projection $p$$p$ corresponds to the map $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$$p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$$z_i\in\F$. Note, that the same formula works for

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$\F=\OO$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JitdNK$$d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$$p_d$ is a sphere $S^{d-1}$$S^{d-1}$. Check directly that the Hopf maps are locally trivial, thus fibrations. \end{zad} \begin{zad} The natural action of

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$GL(n,\F)$ (resp.

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$O(n,\F)$) on

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$\F^n$ induces an action on

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$G_k(\F^n)$. Show that the actions are transitive and describe the isotropy groups (in particular of the canonical subspace $F^k\subset F^n$$F^k\subset F^n$)

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\begin{zad} Prove that there is a free action of the group

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$O(k,\F)$ on

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$V_k(\F^n)$ sucht that the orbit space is homeomorphic to

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$G_k(\F^n)$. Similarly for the noncompact Stiefel manifold.

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\begin{zad} Prove that the map

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$p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal

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$O(k,\F)$-bundle), thus a fibration.

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They are examples of coadjoint orbits [Kirillov2004]

[Milnor&Stasheff1974]

Theorem 1.2.

2 Construction and examples

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3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

• [Kirillov2004] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004. MR2069175 (2005c:22001) Zbl 02121486
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504