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

By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

Algebraic topology is a easy a part of smooth arithmetic, and a few wisdom of this sector is critical for any complex paintings when it comes to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This e-book presents a close remedy 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 technique displays the large inner advancements inside algebraic topology over the last a number of many years, so much of that are mostly unknown to mathematicians in different fields. yet he additionally keeps the classical displays of assorted themes the place applicable. so much chapters finish with difficulties that extra discover and refine the techniques provided. the ultimate 4 chapters supply sketches of considerable components of algebraic topology which are more often than not passed over from introductory texts, and the booklet concludes with a listing of advised readings for these drawn to delving additional into the field.

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The weak Hausdorff property admits a similar characterization. Lemma. If X is a k-space, then X is weak Hausdorff if and only if ∆X is closed in X × X. 2. The category of compactly generated spaces One major source of point-set level pathology can be passage to quotient spaces. Use of compactly generated topologies alleviates this. Proposition. If X is compactly generated and π : X −→ Y is a quotient map, then Y is compactly generated if and only if (π × π)−1 (∆Y ) is closed in X × X. The interpretation is that a quotient space of a compactly generated space by a “closed equivalence relation” is compactly generated.

Then π ◦ ν = id and id ≃ ν ◦ π since we can define a deformation h : N f × I −→ N f of N f onto ν(X) by setting h(x, χ)(t) = (x, χt ), where χt (s) = χ((1 − t)s). We check directly that ρ : N f −→ Y satisfies the CHP. Consider a test diagram A i0  x A×I g ˜ h x x h G Nf xY ρ  G Y. 4. A CRITERION FOR A MAP TO BE A FIBRATION 51 ˜ that makes the We are given g and h such that h ◦ i0 = ρ ◦ g and must construct h diagram commute. We write g(a) = (g1 (a), g2 (a)) and set ˜ t) = (g1 (a), j(a, t)), h(a, where j(a, t)(s) = g2 (a)(s + st) if 0 ≤ s ≤ 1/(1 + t) h(a, s + ts − 1) if 1/(1 + t) ≤ s ≤ 1.

If G has k generators, then χ(B) = 1 − k. If [G : H] = n, then Fb has cardinality n and χ(E) = nχ(B). Therefore 1 − χ(E) = 1 − n + nk. We can extend the idea to realize any group as the fundamental group of some connected space. Theorem. For any group G, there is a connected space X such that π1 (X) is isomorphic to G. 38 GRAPHS Proof. We may write G = F/N for some free group F and normal subgroup N . As above, we may realize the inclusion of N in F by passage to fundamental groups from a cover p : E −→ B.

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