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

By J. P. May

Algebraic topology is a easy a part of sleek arithmetic, and a few wisdom of this region is imperative for any complex paintings with regards to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This e-book presents an in depth remedy of algebraic topology either for academics 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 procedure displays the big inner advancements inside algebraic topology during the last a number of many years, 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 acceptable. so much chapters finish with difficulties that additional discover and refine the techniques awarded. the ultimate 4 chapters supply sketches of considerable parts of algebraic topology which are as a rule passed over from introductory texts, and the ebook concludes with a listing of advised readings for these attracted to delving extra into the field.

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The set of all such U [f ] is a basis for a topology on E since if U [f ] and U ′ [f ′ ] are two such sets and [g] is in their intersection, then W [g] ⊂ U [f ] ∩ U ′ [f ′ ] for any open set W of B such that p[g] ∈ W ⊂ U ∩ U ′ . For u ∈ U , there is a unique [g] in each U [f ] such that p[g] = u. Thus p maps U [f ] homeomorphically onto U and, if we choose a basepoint u in U , then p−1 (U ) is the disjoint union of those U [f ] such that f ends at u. It only remains to show that E is connected, locally path connected, and simply connected, and the second of these is clear.

We have the functor k : wU −→ U , and we have the forgetful functor j : U −→ wU , which embeds U as a full subcategory of wU . Clearly U (X, kY ) ∼ = wU (jX, Y ) for X ∈ U and Y ∈ wU since the identity map kY −→ Y is continuous and continuity of maps defined on compactly generated spaces is compactly determined. Thus k is right adjoint to j. We can construct colimits and limits of spaces by performing these constructions on sets: they inherit topologies that give them the universal properties of colimits and limits in the classical category of spaces.

X i B 0 ¯ : X ×I −→ Y and We first use that A −→ X is a cofibration to obtain a homotopy h ¯ then use the right-hand pushout to see that h and h induce the required homotopy ˜ h. 2. Mapping cylinders and cofibrations Although the HEP is expressed in terms of general test diagrams, there is a certain universal test diagram. Namely, we can let Y in our original test diagram be the “mapping cylinder” M i ≡ X ∪i (A × I), which is the pushout of i and i0 . Indeed, suppose that we can construct a map r that makes the following diagram commute: i0 G A×I v v vv v vv v{ v i×id i M a i ™r r r || | r | r || r   || G X × I.

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