A Concise Course in Algebraic Topology - download pdf or read online

By J. P. May

Algebraic topology is a simple a part of glossy arithmetic, and a few wisdom of this region is integral for any complicated paintings with regards to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This ebook presents an in depth remedy of algebraic topology either for lecturers 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 procedure displays the big inner advancements inside of algebraic topology over the last a number of a long time, so much of that are mostly unknown to mathematicians in different fields. yet he additionally keeps the classical shows of assorted subject matters the place acceptable. so much chapters finish with difficulties that extra discover and refine the techniques awarded. the ultimate 4 chapters offer sketches of considerable parts of algebraic topology which are mostly passed over from introductory texts, and the publication concludes with an inventory of steered readings for these drawn to delving additional into the field.

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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. We have expressed the definition in a form that should make the following statement clear.

There is a functor E (−) : O(G) −→ Cov(B) that is an equivalence of categories. For each subgroup H of G, the covering p : E (G/H) −→ B has a canonical base object e in its fiber over b such that p(π(E (G/H), e)) = H. Moreover, Fb = G/H as a G-set and, for a G-map α : G/H −→ G/K in O(G), the restriction of E (α) : E (G/H) −→ E (G/K) to fibers over b coincides with α. Proof. The idea is that, up to bijection, StE (G/H) (e) must be the same set for each H, but the nature of its points can differ with H.

Abstractions of these ideas are at the heart of modern axiomatic treatments of homotopical algebra and of the foundations of algebraic K-theory. The theories of cofiber and fiber sequences illustrate an important, but informal, duality theory, known as Eckmann-Hilton duality. It is based on the adjunction between Cartesian products and function spaces. Our standing hypothesis that all spaces in sight are compactly generated allows the theory to be developed without further restrictions on the given spaces.

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