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gptkbp:instanceOf
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gptkb:mathematical_concept
gptkb:topology
gptkb:Lie_group
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gptkbp:appearsIn
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gptkb:topology
differential geometry
complex geometry
Lie group theory
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gptkbp:automorphismGroup
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GL(n,Z) ⋉ T^n
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gptkbp:compact
|
true
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gptkbp:definedIn
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product of n circles
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gptkbp:dimensions
|
n
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gptkbp:fundamentalGroup
|
Z^n
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gptkbp:groupStructure
|
(S^1)^n
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gptkbp:hasConnection
|
true
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gptkbp:hasFlatMetric
|
true
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gptkbp:hasSpecialCase
|
T^1 is the circle
T^2 is the standard torus
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gptkbp:homologyGroup
|
H_k(T^n) = (Z^{n \\choose k})
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gptkbp:homotopyType
|
product of n circles
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|
https://www.w3.org/2000/01/rdf-schema#label
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torus T^n
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gptkbp:isAbelianVariety
|
true (if complex structure given)
|
|
gptkbp:isAlgebraicGroup
|
true
|
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gptkbp:isAlgebraicTorus
|
true (in algebraic geometry)
|
|
gptkbp:isBaseSpaceFor
|
principal circle bundle
|
|
gptkbp:isCommutativeGroup
|
true
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gptkbp:isComplexManifold
|
true
|
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gptkbp:isFiberBundleOver
|
T^{n-1}
|
|
gptkbp:isFlatManifold
|
true
|
|
gptkbp:isGroupVariety
|
true
|
|
gptkbp:isHomogeneousSpace
|
true
|
|
gptkbp:isKählerManifold
|
true
|
|
gptkbp:isManifold
|
true
|
|
gptkbp:isNonAbelian
|
true
|
|
gptkbp:isOrientable
|
true
|
|
gptkbp:isParallelizable
|
true
|
|
gptkbp:isParallelizableManifold
|
true
|
|
gptkbp:isPrincipalHomogeneousSpace
|
true
|
|
gptkbp:isQuotientOf
|
gptkb:R^n_/_Z^n
R^n by Z^n
|
|
gptkbp:isRiemannianManifold
|
true
|
|
gptkbp:isSmoothManifold
|
true
|
|
gptkbp:isSymplecticManifold
|
true
|
|
gptkbp:notation
|
(S^1)^n
T^n
|
|
gptkbp:universalCover
|
R^n
itself
|
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gptkbp:bfsParent
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gptkb:compact_Lie_groups
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gptkbp:bfsLayer
|
6
|