By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

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

**Example text**

Proof. If g exists, its properties directly imply that im(f ) ⊂ im(p). For an object x of X and a map α : x0 −→ x in X , let α ˜ be the unique element of St(e0 ) such that p(˜ α) = f (α). If g exists, g(α) must be α ˜ and therefore g(x) must be the target T (f (α))(e0 ) of α ˜ . The inclusion f (π(X , x0 )) ⊂ p(π(E , e0 )) ensures that T (f (α))(e0 ) is independent of the choice of α, so that g so specified is a well defined functor. In fact, given another map α′ : x0 −→ x, α−1 ◦ α′ is an element of π(X , x0 ).

Clearly g ◦ j = i ◦ e. Using that the pairs (I × I, I × {0}) and (I × I, I × {0} ∪ ∂I × I) are homeomorphic, we can construct a homotopy between homotopies Λ by applying HEP to the diagram ⊂ G A×I ×I tt Γ tt t t tt tz t i×id i×id TX lll dtttt l l l γ ll ttΛ l tt lll tt lll G (X × I × 0) ∪ (X × ∂I × I) ⊂ X × I × I. (A × I × 0) ∪ (A × ∂I × I) Here ψ(m(x, 2s)) if s ≤ 1/2 n(x, 2s − 1) if s ≥ 1/2, γ(x, 0, t) = (g ◦ f )(x) = (ψ ◦ g ′ ◦ f )(x), γ(x, s, 0) = and γ(x, 1, t) = x, while i(h(a, 2s/(1 + t))) if 2s ≤ 1 + t i(a) if 2s ≥ 1 + t Define H(x, s) = Λ(x, s, 1).

There exists a map g : E −→ E ′ of coverings with g(e) = e′ if and only if p∗ (π1 (E, e)) ⊂ p′∗ (π1 (E ′ , e′ )), and there is then only one such g. In particular, two maps of covers g, g ′ : E −→ E ′ coincide if g(e) = g ′ (e) for any one e ∈ E. Moreover, g is a homeomorphism if and only if the displayed inclusion of subgroups of π1 (B, b) is an equality. Therefore E and E ′ are homeomorphic if and only if p∗ (π1 (E, e)) and p′∗ (π1 (E ′ , e′ )) are conjugate whenever p(e) = p′ (e′ ). Corollary.

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

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