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. They play a key role in topology and geometry as the universal spaces of vector bundles.

2 Construction and examples

Let $\F=\Rr ,\Cc , \Hh$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\newcommand{\F}{\mathbb{F}}\F=\Rr ,\Cc , \Hh$ be the real, complex or quaternion field and $V$$V$ a vector space over $\F/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_0rLgXR$$\F$ of dimension $n$$n$ and let $k\leq n$$k\leq n$. A Grassmann manifolds of $k$$k$-dimensional subspaces is a set $G_k(V)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_6kM4Ca$$G_k(V)$ of $k$$k$-dimensional subspaces. The set $G_k(V)$$G_k(V)$ is a quotient of a subset of $V\times ...\times V/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qbO8It$$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 2.1 [Milnor&Stasheff1974]. $G_k(V)$$G_k(V)$ is a Hausdorff, compact space.

Theorem 2.2 [Milnor&Stasheff1974]. $G_k(V)$$G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_20czgN$$dk(n-k)$. For $\F=\Cc ,\Hh$$\F=\Cc ,\Hh$ it is also a complex manifold.

Note that the Grassmann manifold $G_k(V)$$G_k(V)$ around $W\in G_k(V)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_2rWFf7$$W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_QzArDr$$Hom (W^\bot ,W).$

Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_KvFqpM$$GL(V)$ acts transitively on $G_k(V)$$G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$$V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_K5G0z7$$O(V)$ acts transitively and the isotropy group of $W$$W$ is $O(W^\bot)\times O(W)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_gYXA9s$$O(W^\bot)\times O(W)$.

The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_eOo37O$$\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_WoNXub$$G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_gihmhy$$E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ The total space of the associated principal bundle is a Stiefel manifold.

The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$$TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$

Proposition 2.3. There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_czCFtV$$G_k(V)\simeq G_{n-k}(V^*)$.

The Grassmannians $G_1(V)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_o2M54i$$G_1(V)$ are projective spaces, denoted $P (V)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_CpGP5G$$P (V)$. Note that $G_1(F^2)=S^d/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_oM4Mu5$$G_1(F^2)=S^d$, where $d=dim_R F/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_iS5Jiu$$d=dim_R F$. If we identify $S^d/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_WzAkvT$$S^d$ with the one-point compactification of $\F$$\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Ujks8i$$p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_0YJYaJ$$p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_SLGVC9$$z_i\in\F$. Note, that the same formula works for octonions $\Oo$$\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$$d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_oupfuA$$p_d$ is a sphere $S^{d-1}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_AUV8K1$$S^{d-1}$.

There is an embedding of the Grassmannian $G_k(V)$$G_k(V)$ in the Cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_4opbrt$$\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subspace the orthogonal projection on it. If $V$$V$ is equipped with a norm, the embedding defines a natural (operator) metric on $G_k(V)$$G_k(V)$.

Infinite dimensional Grassmannians. Natural inclusions of vector space $\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$$\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$ defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted $G_k(\F^{\infty} )$$G_k(\F^{\infty} )$ and also $BGL(k,\F)$$BGL(k,\F)$. One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions $G_1(\F^2)\subset G_2(F^4)\subset ... \subset G_n(F^2n) \subset ...$$G_1(\F^2)\subset G_2(F^4)\subset ... \subset G_n(F^2n) \subset ...$ and its colimit is denoted $B\GL (\F).$$B\GL (\F).$

Invariants

Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups $\pi_i(G_k (V))$$\pi_i(G_k (V))$ do not depend on $V$$V$, if $k\leq\leq dim V.$$k\leq\leq dim V.$ Homotopy groups in the stable range are described by the Bott periodicity theorem.

Proposition 4.1 {(R.Bott).} For each $i>0$$i>0$ there are isomorphisms: $\pi_i(BGL(\Rr) \simeq pi_{i+8}(BGL(\Rr)$$\pi_i(BGL(\Rr) \simeq pi_{i+8}(BGL(\Rr)$ and $\pi_i(BGL(\Cc) \simeq pi_{i+2}(BGL(\Cc)$$\pi_i(BGL(\Cc) \simeq pi_{i+2}(BGL(\Cc)$

Bot periodicity, relation to homotopy group of spheres

Cohomology groups.

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3 Classification/Characterization

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4 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

5 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