By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

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**Additional resources for A Concise Course in Algebraic Topology**

**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).

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

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