By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

Algebraic topology is a easy a part of glossy arithmetic, and a few wisdom of this zone is vital for any complicated paintings in terms of geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This ebook offers an in depth therapy of algebraic topology either for academics of the topic and for complicated graduate scholars in arithmetic both focusing on this sector or carrying on with directly to different fields. J. Peter May's technique displays the large inner advancements inside of algebraic topology over the last a number of a long time, so much of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical displays of assorted issues the place applicable. such a lot chapters finish with difficulties that extra discover and refine the recommendations awarded. the ultimate 4 chapters offer sketches of considerable parts of algebraic topology which are quite often passed over from introductory texts, and the booklet concludes with a listing of steered readings for these drawn to delving additional into the field.

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Extra info for A Concise Course in Algebraic Topology

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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.

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

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