By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

**Read Online or Download A Concise Course in Algebraic Topology PDF**

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**Extra info for A Concise Course in Algebraic Topology**

**Example text**

Choose a base object b of B and let G = π(B, b). There is a functor E (−) : O(G) −→ Cov(B) that is an equivalence of categories. For each subgroup H of G, the covering p : E (G/H) −→ B has a canonical base object e in its fiber over b such that p(π(E (G/H), e)) = H. Moreover, Fb = G/H as a G-set and, for a G-map α : G/H −→ G/K in O(G), the restriction of E (α) : E (G/H) −→ E (G/K) to fibers over b coincides with α. Proof. The idea is that, up to bijection, StE (G/H) (e) must be the same set for each H, but the nature of its points can differ with H.

If p : E −→ B is a covering, then p is a fibration with a unique path lifting function s. Proof. The unique lifts of paths with a given initial point specify s. Fibrations and cofibrations are related by the following useful observation. Lemma. If i : A −→ X is a cofibration and B is a space, then the induced map p = B i : B X −→ B A is a fibration. Proof. It is an easy matter to check that we have a homeomorphism B Mi = B X×{0}∪A×I ∼ = B X ×p (B A )I = N p. If r : X × I −→ M i is a retraction, then Br : N p ∼ = (B X )I = B Mi −→ B X×I ∼ is a path lifting function.

Clearly φ = φ(¯ n), and it is easy to check that this bijection between W H and AutG (S) is an isomorphism of groups. We shall also need to consider G-maps between different G-sets G/H. Lemma. A G-map α : G/H −→ G/K has the form α(gH) = gγK, where the element γ ∈ G satisfies γ −1 hγ ∈ K for all h ∈ H. Proof. If α(eH) = γK, then the relation γK = α(eH) = α(hH) = hα(eH) = hγK implies that γ −1 hγ ∈ K for h ∈ H. Definition. The category O(G) of canonical orbits has objects the G-sets G/H and morphisms the G-maps of G-sets.

### A Concise Course in Algebraic Topology by J. P. May

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