By J. P. May

Algebraic topology is a uncomplicated a part of smooth arithmetic, and a few wisdom of this zone is essential for any complicated paintings in terms of geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This e-book offers an in depth remedy of algebraic topology either for lecturers of the topic and for complex graduate scholars in arithmetic both focusing on this quarter or carrying on with directly to different fields. J. Peter May's process displays the big inner advancements inside algebraic topology over the last numerous many years, such a lot of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical shows of assorted themes the place applicable. so much chapters finish with difficulties that additional discover and refine the options awarded. 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 e-book concludes with an inventory of urged readings for these attracted to delving extra into the sector.

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C) An open subset U of a compactly generated space X is compactly generated if each point has an open neighborhood in X with closure contained in U . (2) * A Tychonoff (or completely regular) space X is a T1 -space (points are closed) such that for each point x ∈ X and each closed subset A such that x∈ / A, there is a function f : X −→ I such that f (x) = 0 and f (a) = 1 if a ∈ A. , Kelley, General Topology). (a) A space is Tychonoff if and only if it can be embedded in a cube (a product of copies of I).

Theorem. A subgroup H of a free group G is free. If G is free on k generators and H has finite index n in G, then H is free on 1 − n + nk generators. Proof. Realize G as π1 (B), where B is the wedge of one circle for each generator of G in a given free basis. Construct a covering p : E −→ B such that p∗ (π1 (E)) = H. Since E is a graph, H must be free. 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.

It is easy to check that p : E (G/H) −→ B is a covering, and it is clear that p(π(E (G/H), e)) = H. This defines the object function of the functor E : O(G) −→ Cov(B). To define E on morphisms, consider α : G/H −→ G/K. If α(eH) = gK, then g −1 Hg ⊂ K and α(f H) = f gK. The functor E (α) : E (G/H) −→ E (G/K) sends the object f H to the object α(f H) = f gK and sends the morphism f ◦ h ◦ f −1 to the same morphism of B regarded as f g ◦ g −1 hg ◦ g −1 f −1 . It is easily checked that each E (α) is a well defined functor, and that E is functorial in α.