Download e-book for kindle: A Concise Course in Algebraic Topology by J. P. May

By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

Algebraic topology is a simple a part of glossy arithmetic, and a few wisdom of this quarter is integral for any complicated paintings when it comes to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This booklet presents a close therapy of algebraic topology either for academics 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 method displays the large inner advancements inside of algebraic topology over the last numerous a long time, such a lot of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical displays of assorted subject matters the place acceptable. such a lot chapters finish with difficulties that extra discover and refine the recommendations provided. 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 an inventory of prompt readings for these attracted to delving extra into the field.

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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.

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