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

By J. P. May

Algebraic topology is a easy a part of smooth arithmetic, and a few wisdom of this region is necessary for any complicated paintings when it comes to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This ebook presents a close remedy of algebraic topology either for lecturers of the topic and for complex graduate scholars in arithmetic both focusing on this zone or carrying on with directly to different fields. J. Peter May's method displays the large inner advancements inside 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 varied themes the place applicable. such a lot chapters finish with difficulties that extra discover and refine the thoughts awarded. the ultimate 4 chapters offer sketches of considerable components of algebraic topology which are commonly passed over from introductory texts, and the ebook concludes with an inventory of instructed readings for these attracted to delving extra into the field.

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It is universal if p(π(E , e)) = {e}. Clearly a covering space is regular or universal if and only if its associated covering of fundamental groupoids is regular or universal. A covering of groupoids is universal if and only if π(B, b) acts freely on Fb , and then Fb is isomorphic to π(B, b) as a π(B, b)-set. Specializing to covering spaces, this sharpens our earlier claim that the elements of Fb and π1 (B, b) are in bijective correspondence. 5. The classification of coverings of groupoids Fix a small connected groupoid B throughout this section and the next.

Show that the space of G-maps G/H −→ X is naturally homeomorphic to X H . In particular, O(G/H, G/K) ∼ = (G/K)H . 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.

For (ii), there is a map g : e −→ e′ since E is connected. Conjugation by g gives a homomorphism π(E , e) −→ π(E , e′ ) that maps under p to conjugation of π(B, b) by its element p(g). For (iii), the surjectivity of p on St(e) gives that any f ∈ π(B, b) is of the form p(g) for some g ∈ St(e). If e′ is the target of g, then p(π(E , e′ )) is the conjugate of p(π(E , e)) by f . The fibers Fb of a covering of groupoids are related by translation functions. Definition. Let p : E −→ B be a covering of groupoids.

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