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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:basePointIndependence
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up to isomorphism if space is path-connected
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gptkbp:capturedBy
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information about holes in a space
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gptkbp:category
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gptkb:topology
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gptkbp:dependsOn
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base point
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gptkbp:describes
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loops up to homotopy
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gptkbp:exampleFor
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gptkb:butter
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gptkbp:field
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gptkb:topology
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gptkbp:generalizes
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first homotopy group
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gptkbp:hasInvariant
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gptkb:homotopy_equivalence
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gptkbp:heldBy
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gptkb:group_of_people
gptkb:topology
first homotopy group
functor from Top_* to Grp
special case of homotopy group
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|
https://www.w3.org/2000/01/rdf-schema#label
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Fundamental group
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gptkbp:identityElement
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constant loop
reverse loop
|
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gptkbp:introduced
|
gptkb:Henri_Poincaré
|
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gptkbp:introducedIn
|
1895
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gptkbp:isFunctorial
|
yes
|
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gptkbp:isNonAbelian
|
not always
|
|
gptkbp:notation
|
π₁(X, x₀)
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|
gptkbp:operator
|
concatenation of loops
|
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gptkbp:relatedTo
|
gptkb:Seifert–van_Kampen_theorem
gptkb:fundamental_groupoid
homotopy theory
loop space
universal covering space
homology group
higher homotopy groups
covering spaces
path-connectedness
deck transformation group
homotopy lifting property
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|
gptkbp:symbol
|
π₁(X, x₀)
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gptkbp:trivialFor
|
simply connected spaces
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gptkbp:usedFor
|
classifying topological spaces
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gptkbp:usedIn
|
gptkb:algebraic_geometry
differential geometry
group theory
mathematical physics
knot theory
|
|
gptkbp:π₁(Projective_plane)
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gptkb:Z/2Z
|
|
gptkbp:π₁(R²)
|
trivial group
|
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gptkbp:π₁(Sphere)
|
trivial group
|
|
gptkbp:π₁(S¹)
|
Z
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|
gptkbp:π₁(Torus)
|
Z × Z
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gptkbp:bfsParent
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gptkb:algebraic_geometry
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gptkbp:bfsLayer
|
5
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