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By J. P. May

Algebraic topology is a uncomplicated a part of sleek arithmetic, and a few wisdom of this quarter is quintessential for any complicated paintings in relation 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 region or carrying on with directly to different fields. J. Peter May's technique displays the large inner advancements inside of algebraic topology over the last a number of many years, so much of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical shows of varied subject matters the place applicable. so much chapters finish with difficulties that extra discover and refine the recommendations offered. the ultimate 4 chapters offer sketches of considerable components of algebraic topology which are commonly passed over from introductory texts, and the e-book concludes with an inventory of instructed readings for these attracted to delving additional into the field.

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A subspace A of X is said to be “compactly closed” if g −1 (A) is closed in K for any map g : K −→ X from a compact space K into X. When X is weak Hausdorff, this holds if and only if the intersection of A with each compact subset of X is closed. A space X is a “k-space” if every compactly closed subspace is closed. A space X is “compactly generated” if it is a weak Hausdorff k-space. For example, any locally compact space and any weak Hausdorff space that satisfies the first axiom of countability (every point has a countable neighborhood basis) is compactly generated.

8. THE CONSTRUCTION OF COVERINGS OF SPACES 33 6. Let X be a G-space. Show that passage to fixed point spaces, G/H −→ X H , is the object function of a contravariant functor X (−) : O(G) −→ U . CHAPTER 4 Graphs We define graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space. 1. The definition of graphs We give the definition in a form that will later make it clear that a graph is exactly a one-dimensional CW complex.

Then passage to orbit spaces defines a functor X/(−) : O(G) −→ U . Proof. The functor sends G/H to X/H and sends a map α : G/H −→ G/K to the map X/H −→ X/K that sends the coset Hx to the coset Kγ −1 x, where α is given by the subconjugacy relation γ −1 Hγ ⊂ K. The starting point of the construction of general covers is the following description of regular covers and in particular of the universal cover. Proposition. Let p : E −→ B be a cover such that Aut(E) acts transitively on Fb . Then the cover p is regular and E/ Aut(E) is homeomorphic to B.

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