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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:application
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gptkb:Obstruction_theory
Classifying topological spaces
Homotopy classification of maps
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gptkbp:category
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Algebraic invariants
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gptkbp:definedIn
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Pointed topological spaces
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gptkbp:defines
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π_n(X, x_0) = homotopy classes of based maps from S^n to X
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gptkbp:describes
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Classes of maps from n-spheres to a space
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gptkbp:example
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π_1(S^1) = ℤ
π_2(S^2) = ℤ
π_3(S^2) = ℤ
π_n(S^n) = ℤ
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gptkbp:field
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gptkb:Algebraic_topology
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gptkbp:firstHomotopyGroup
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gptkb:Fundamental_group
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gptkbp:generalizes
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gptkb:Fundamental_group
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gptkbp:higherHomotopyGroups
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π_n(X) for n > 1
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https://www.w3.org/2000/01/rdf-schema#label
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Homotopy group
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gptkbp:introduced
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gptkb:Henri_Poincaré
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gptkbp:notation
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π_n(X)
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gptkbp:property
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π_1(X) may be non-abelian
π_n(X) is abelian for n > 1
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gptkbp:relatedTo
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gptkb:Homotopy_category
gptkb:Freudenthal_suspension_theorem
gptkb:Postnikov_system
gptkb:Whitehead_theorem
gptkb:Eilenberg–MacLane_space
gptkb:Hurewicz_theorem
gptkb:Cohomology_group
gptkb:Homotopy_colimit
gptkb:Homotopy_equivalence
gptkb:Homotopy_fiber
gptkb:Homotopy_lifting_property
gptkb:Homotopy_limit
gptkb:Homotopy_type
gptkb:Long_exact_sequence_of_homotopy_groups
gptkb:Loop_space
gptkb:Simplicial_set
gptkb:Stable_homotopy_group
gptkb:Homology_group
CW complex
Spectral sequence
Homotopy extension property
Homotopy pullback
Homotopy pushout
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gptkbp:studies
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gptkb:Topological_spaces
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gptkbp:type
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gptkb:group_of_people
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gptkbp:usedIn
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gptkb:Algebraic_topology
Homotopy theory
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gptkbp:bfsParent
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gptkb:Obstruction_theory
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gptkbp:bfsLayer
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7
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