Grassmann manifolds

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== Introduction ==
== Introduction ==
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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. See also [[Wikipedia:Grassmannian|Grassmannian]]
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. See also [[Wikiepdia:Grassmannian|Grassmannian]]
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== Construction and examples ==
== Construction and examples ==
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Let $\F=\Rr ,\Cc , \Hh$ be the real, complex or quaternion field and $V$ a vector space over $\F$ of dimension $n$ and let $k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology.
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=== Construction ===
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{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}}
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Let $\F=\Rr ,\Cc , \Hh$ be the real, complex or quaternion field and $V$ a vector space over $\F$ of dimension $n$ and let $k\leq n$. A Grassmannian of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group $\GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$.
{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}}
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{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact, connected smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}}
Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$
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Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$
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Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$.
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{{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}}
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=== The canonical bundle ===
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The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ The total space of the associated principal bundle is a Stiefel manifold.
The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ 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: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$
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The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$</wikitex>
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=== Low dimensional Grassmannians ===
{{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}}
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The Grassmannians $G_1(V)$ are projective spaces, denoted $P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_{\Rr} F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions $\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d$ is a sphere $S^{d-1}$.
The Grassmannians $G_1(V)$ are projective spaces, denoted $P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_R F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions $\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d$ is a sphere $S^{d-1}$.
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=== Embeddings of Grassmannians into affine and projective space ===
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There is an embedding of the Grassmannian $G_k(V)$ in the Cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subspace the orthogonal projection on it. If $V$ is equipped with a norm, the embedding defines a natural (operator) metric on $G_k(V)$.
There is an embedding of the Grassmannian $G_k(V)$ in the Cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subspace the orthogonal projection on it. If $V$ is equipped with a norm, the embedding defines a natural (operator) metric on $G_k(V)$.
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Infinite dimensional Grassmannians. Natural inclusions of vector space $\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$ defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted $G_k(\F^{\infty} )$ and also $BGL(k,\F)$. One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions $G_1(\F^2)\subset G_2(F^4)\subset ... \subset G_n(F^2n) \subset ...$ and its colimit is denoted $B\GL (\F).$
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=== Infinite dimensional Grassmannians ===
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Infinite dimensional Grassmannians. Natural inclusions of vector space $\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$ defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted $G_k(\F^{\infty} )$ and also $BGL(k,\F)$. One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions $G_1(\F^2)\subset G_2(\F^4)\subset ... \subset G_n(\F^{2n}) \subset ...$ and its colimit is denoted $B\GL (\F).$
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== Invariants ==
== Invariants ==
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=== Homotopy groups ===
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Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups $\pi_i(G_k (V))$ do not depend on $V$, if $k\leq\leq dim V.$ Homotopy groups in the stable range are described by the Bott periodicity theorem.
Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups $\pi_i(G_k (V))$ do not depend on $V$, if $k\leq\leq dim V.$ Homotopy groups in the stable range are described by the Bott periodicity theorem.
{{beginthm|Proposition|{(R.Bott)}}} For each $i>0$ there are isomorphisms: $\pi_i(BGL(\Rr) \simeq \pi_{i+8}(BGL(\Rr)$ and $\pi_i(BGL(\Cc) \simeq \pi_{i+2}(BGL(\Cc)$
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{{beginthm|Proposition|{(R.Bott)}}} For each $i>0$ there are isomorphisms: $\pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr))$ and $\pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))$
{{endthm}}
{{endthm}}
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Cohomology groups.
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=== Cohomology groups ===
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== Further discussion ==
== Further discussion ==
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Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}.
Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}.
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Latest revision as of 18:05, 12 October 2013

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

[edit] 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. See also Grassmannian

[edit] 2 Construction and examples

[edit] 2.1 Construction

Let \F=\Rr ,\Cc , \Hh be the real, complex or quaternion field and V a vector space over \F of dimension n and let k\leq n. A Grassmannian of k-dimensional subspaces is a set G_k(V) of k-dimensional subspaces. The set G_k(V) is a quotient of a subset of V\times ...\times V consisting of linearly independent k-tuples of vectors with the subspace topology. We define topology on G_k(V) as the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group \GL(V) acts transitively on G_k(V) with an isotropy group consisting of automorphisms preserving a given subspace. If the space V is equipped with a scalar product (hermitian metric resp.) then the group of isometries O(V) acts transitively and the isotropy group of W is O(W^\bot)\times O(W).

Theorem 2.1 [Milnor&Stasheff1974]. G_k(V) is a Hausdorff, compact, connected smooth manifold of dimension dk(n-k). For \F=\Cc ,\Hh it is also a complex manifold.

Note that the Grassmann manifold G_k(V) around W\in G_k(V) is locally modelled on the vector space Hom (W^\bot ,W).

Proposition 2.2. There exist a natural diffeomorphism G_k(V)\simeq G_{n-k}(V^*).

[edit] 2.2 The canonical bundle

The Grassmann manifold is equipped with the canonical, tautological vector bundle \gamma^V_k. which is a subbundle of the trivial bundle 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 \} 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).

[edit] 2.3 Low dimensional Grassmannians

The Grassmannians G_1(V) are projective spaces, denoted P (V). Note that G_1(F^2)=S^d, where d=dim_{\Rr} F. If we identify S^d with the one-point compactification of \F the projection of the canonical principal bundle corresponds to the map p_d :S^{2d-1}\to S^d given by p_d(z_0,z_1)=z_0/z_1 where z_i\in\F. Note, that the same formula works for octonions \Oo, however the higher dimensional projective spaces over octonions do not exist. The maps p_d :S^{2d-1}\to S^d for d= 2,4,8 are called the Hopf maps and they play a very important role in homotopy theory; a fibre of p_d is a sphere S^{d-1}.

[edit] 2.4 Embeddings of Grassmannians into affine and projective space

There is an embedding of the Grassmannian G_k(V) in the Cartesian space \F^{n^2}=\Hom\,(F^n,F^n) which assigns to every subspace the orthogonal projection on it. If V is equipped with a norm, the embedding defines a natural (operator) metric on G_k(V).

[edit] 2.5 Infinite dimensional Grassmannians

Infinite dimensional Grassmannians. Natural inclusions of vector space \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} ) and also 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 ... and its colimit is denoted B\GL (\F).

[edit] 3 Invariants


[edit] 3.1 Homotopy groups

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)) do not depend on V, if k\leq\leq dim V. Homotopy groups in the stable range are described by the Bott periodicity theorem.

Proposition 3.1 {(R.Bott).} For each i>0 there are isomorphisms: \pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr)) and \pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))

[edit] 3.2 Cohomology groups

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[edit] 4 Classification/Characterization

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[edit] 5 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

[edit] 6 References

[edit] 7 External links

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