Grassmann manifolds
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== Construction and examples == | == Construction and examples == | ||
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=== Construction === | === Construction === | ||
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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 $O(W^\bot)\times O(W)$. | + | 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 Hausdorff, compact, connected smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}} | {{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}} | ||
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{{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}} | {{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}} | ||
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=== The canonical bundle === | === 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: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$ | + | 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 === | === Low dimensional Grassmannians === | ||
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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= | + | 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 $S^{d-1}$. |
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=== Embeddings of Grassmannians into affine and projective space === | === 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 === | === 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 $B\GL (\F).$ | + | 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 $\pi_i(BGL(\Cc) \simeq \pi_{i+2}(BGL(\Cc)$ | + | {{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 | + | === 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 19:05, 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
Tex syntax errora vector space over of dimension and let . A Grassmannian of -dimensional subspaces is a set of -dimensional subspaces. The set is a quotient of a subset of consisting of linearly independent -tuples of vectors with the subspace topology. We define topology on as the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group acts transitively on with an isotropy group consisting of automorphisms preserving a given subspace. If the space
Tex syntax erroris equipped with a scalar product (hermitian metric resp.) then the group of isometries acts transitively and the isotropy group of is .
Note that the Grassmann manifold around is locally modelled on the vector space
[edit] 2.2 The canonical bundle
The Grassmann manifold is equipped with the canonical, tautological vector bundle which is a subbundle of the trivial bundle . The total space is 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:
[edit] 2.3 Low dimensional Grassmannians
The Grassmannians are projective spaces, denoted . Note that , where . If we identify with the one-point compactification of the projection of the canonical principal bundle corresponds to the map given by where . Note, that the same formula works for octonions , however the higher dimensional projective spaces over octonions do not exist. The maps for are called the Hopf maps and they play a very important role in homotopy theory; a fibre of is a sphere .
[edit] 2.4 Embeddings of Grassmannians into affine and projective space
Tex syntax erroris equipped with a norm, the embedding defines a natural (operator) metric on .
[edit] 2.5 Infinite dimensional Grassmannians
Infinite dimensional Grassmannians. Natural inclusions of vector space defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted and also . One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions and its colimit is denoted
[edit] 3 Invariants
[edit] 3.1 Homotopy groups
Tex syntax error, if Homotopy groups in the stable range are described by the Bott periodicity theorem.
Proposition 3.1 {(R.Bott).} For each there are isomorphisms: and
[edit] 3.2 Cohomology groups
...
[edit] 4 Classification/Characterization
...
[edit] 5 Further discussion
Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].
[edit] 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
[edit] 7 External links
- The Wikipedia page on Grassmannian