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... is, what the product map π2 (X) × π2 (X) → π2 (X) is, and what the inverse map π2 (X) → π2 (X) is.) Prove that the group π2 (X) is commutative. Solution: π2 (X) is the set of homotopy classes of maps f : [0, 1]2 → X that send ∂[0, 1]2 to the base point of X, where the homotopies are taken relatively ...
... is, what the product map π2 (X) × π2 (X) → π2 (X) is, and what the inverse map π2 (X) → π2 (X) is.) Prove that the group π2 (X) is commutative. Solution: π2 (X) is the set of homotopy classes of maps f : [0, 1]2 → X that send ∂[0, 1]2 to the base point of X, where the homotopies are taken relatively ...
Homework sheet 4
... 1. Recall that a topological space is called irreducible iff it cannot be written as the disjoint union of two proper closed subsets. (a) Prove that a topological space X is irreducible iff any two nonempty open subsets of X have non-empty intersection. (b) Prove that if a topological space X is the ...
... 1. Recall that a topological space is called irreducible iff it cannot be written as the disjoint union of two proper closed subsets. (a) Prove that a topological space X is irreducible iff any two nonempty open subsets of X have non-empty intersection. (b) Prove that if a topological space X is the ...
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... Let us recall the definition of a topological group; this is a group (G, ., e) together with a topology on G such that (x, y) 7→ xy −1 is continuous, i.e., from G × G into G. Note also that G × G is regarded as a topological space defined by the product topology. Definition 0.1. Consider G to be a t ...
... Let us recall the definition of a topological group; this is a group (G, ., e) together with a topology on G such that (x, y) 7→ xy −1 is continuous, i.e., from G × G into G. Note also that G × G is regarded as a topological space defined by the product topology. Definition 0.1. Consider G to be a t ...
