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

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 quarter is vital for any complex paintings on the subject of geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This publication offers a close remedy of algebraic topology either for academics of the topic and for complex graduate scholars in arithmetic both focusing on this zone or carrying on with directly to different fields. J. Peter May's strategy displays the large inner advancements inside algebraic topology during the last a number of many years, such a lot of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical shows of varied subject matters the place acceptable. such a lot chapters finish with difficulties that additional discover and refine the suggestions awarded. the ultimate 4 chapters offer sketches of considerable components of algebraic topology which are mostly passed over from introductory texts, and the e-book concludes with an inventory of prompt readings for these drawn to delving extra into the field.

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Example text

A criterion for a map to be a cofibration We want a criterion that allows us to recognize cofibrations when we see them. We shall often consider pairs (X, A) consisting of a space X and a subspace A. Cofibration pairs will be those pairs that “behave homologically” just like the associated quotient spaces X/A. Definition. A pair (X, A) is an NDR-pair (= neighborhood deformation retract pair) if there is a map u : X −→ I such that u−1 (0) = A and a homotopy h : X × I −→ X such that h0 = id, h(a, t) = a for a ∈ A and t ∈ I, and h(x, 1) ∈ A if u(x) < 1; (X, A) is a DR-pair if u(x) < 1 for all x ∈ X, in which case A is a deformation retract of X.

By definition, f e is the target of the map f˜ ∈ StE (e) such that p(f˜) = f . Clearly g(f e) is the target of g(f˜) ∈ StE ′ (g(e)) and p′ (g(f˜)) = p(f˜) = f . Again by definition, this gives g(f e) = f g(e). The previous theorem shows that restriction to fibers is an injection on Cov(E , E ′ ). To show surjectivity, let α : Fb −→ Fb′ be a G-map. Choose e ∈ Fb and let e′ = α(e). Since α is a G-map, the isotropy group p(π(E , e)) of e is contained in the isotropy group p′ (π(E ′ , e′ )) of e′ .

We begin with the following result, which deserves to be called the fundamental theorem of covering space theory and has many other applications. It asserts that the fundamental group gives the only “obstruction” to solving a certain lifting problem. Recall our standing assumption that all given spaces are connected and locally path connected. Theorem. Let p : E −→ B be a covering and let f : X −→ B be a continuous map. Choose x ∈ X, let b = f (x), and choose e ∈ Fb . There exists a map g : X −→ E such that g(x) = e and p ◦ g = f if and only if f∗ (π1 (X, x)) ⊂ p∗ (π1 (E, e)) in π1 (B, b).

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


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