Algebraic Topology: Oaxtepec 1991 by Tangora M.C. (ed.)

By Tangora M.C. (ed.)

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1. The rotation group O(n + 1) acts on S n . Show that for any M ∈ O(n + 1) the induced morphism M : Hn (S n ) → Hn (S n ) is the multiplication by det M = ±1. 2. Let a : S n → S n be the antipodal map, a(x) = −x. Show that a : Hn (S n ) → Hn (S n ) is the multiplication by (−1)n+1 . 10. We show that the inclusion map (En+ , S n−1 ) → (S n , En− ) is an excision. 4 are not satisfied. However it is enough to consider the subspace V = x ∈ S n | x0 > − 21 . V can be excised from (S n , En− ). But (En+ , S n−1 ) is a deformation retract of (S n − V, En− − V ) so that we are done.

It can be described in another way by the following construction. 12. The set P, endowed with the topology whose base of open subsets consists of the sets s(U ) for U open in X and s ∈ P(U ), is called the ´etal´e space of the presheaf P. 2A function is locally constant on U if it is constant on any connected component of U . 1. 13. , every point u ∈ P has an open neighbourhood U such that π : U → π(U ) is a homeomorphism. (2) Show that for every open set U ⊂ X and every s ∈ P(U ), the section s : U → P is continuous.

The proof of the following Lemma is an elementary computation. 6. Let σ =< E0 , . . , Ek >, with E0 , . . , Ek ∈ Rn . The diameter of every simplex in the singular chain Σ(σ) ∈ Sk (Rn ) is at most k/k + 1 times the diameter of σ. 7. Let X be a topological space, U = {Uα } an open cover, and σ a singular k-simplex in X. There is a natural number r > 0 such that every singular simplex in Σr (σ) is contained in a open set Uα . Proof. As ∆k is compact there is a real positive number such that σ maps a neighbourhood of radius of every point of ∆k into some Uα .

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