Read e-book online A Concise Course in Algebraic Topology PDF

By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

Algebraic topology is a uncomplicated a part of smooth arithmetic, and a few wisdom of this sector is imperative for any complex paintings when it comes to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This ebook offers a close remedy of algebraic topology either for lecturers of the topic and for complicated graduate scholars in arithmetic both focusing on this quarter or carrying on with directly to different fields. J. Peter May's technique displays the big inner advancements inside algebraic topology during the last a number of a long time, so much of that are mostly unknown to mathematicians in different fields. yet he additionally keeps the classical shows of assorted issues the place acceptable. so much chapters finish with difficulties that additional discover and refine the innovations offered. the ultimate 4 chapters offer sketches of considerable parts of algebraic topology which are generally passed over from introductory texts, and the e-book concludes with a listing of instructed readings for these attracted to delving additional into the field.

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Therefore E and E ′ are isomorphic if and only if p(π(E , e)) and p′ (π(E ′ , e′ )) are conjugate whenever p(e) = p′ (e′ ). Corollary. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover. That the universal cover does exist will be proved in the next section. It is useful to recast the previous theorem in terms of actions on fibers. Theorem. Let p : E −→ B and p′ : E ′ −→ B be coverings, choose a base object b ∈ B, and let G = π(B, b). If g : E −→ E ′ is a map of coverings, then g restricts to a map Fb −→ Fb′ of G-sets, and restriction to fibers specifies a bijection between Cov(E , E ′ ) and the set of G-maps Fb −→ Fb′ .

Let G act transitively on a set S, choose s ∈ S, and let H = Gs . Then W H is isomorphic to the group AutG (S) of automorphisms of the G-set S. Proof. For n ∈ N H with image n ¯ ∈ W H, define an automorphism φ(¯ n) of S by φ(¯ n)(gs) = gns. For an automorphism φ of S, we have φ(s) = ns for some n ∈ G. For h ∈ H, hns = φ(hs) = φ(s) = ns, hence n−1 hn ∈ Gs = H and n ∈ N H. 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.

This leads to the following adjointness homeomorphism, which holds without restriction when we work in the category of compactly generated spaces. Proposition. For spaces X, Y , and Z in U , the canonical bijection Z (X×Y ) ∼ = (Z Y )X is a homeomorphism. Observe in particular that a homotopy X × I −→ Y can equally well be viewed as a map X −→ Y I . These adjoint, or “dual,” points of view will play an important role in the next two chapters. PROBLEMS (1) (a) Any subspace of a weak Hausdorff space is weak Hausdorff.

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


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