Download e-book for iPad: Concise course in algebraic topology by J. P. May

Algebraic Geometry

By J. P. May

Algebraic topology is a simple a part of sleek arithmetic, and a few wisdom of this sector is quintessential for any complex paintings when it comes 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 academics 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 technique displays the big inner advancements inside algebraic topology over the last numerous many years, so much of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical displays of varied subject matters the place acceptable. so much chapters finish with difficulties that additional discover and refine the suggestions provided. the ultimate 4 chapters offer sketches of considerable parts of algebraic topology which are quite often passed over from introductory texts, and the ebook concludes with a listing of urged readings for these drawn to delving additional into the sphere.

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3. Give an example of a surjective local homeomorphism that is not a covering. 4. * Let f : X −→ Y be a local homeomorphism, where X is compact. ) covering with finite fibers. Let X be a G-space, where G is a (discrete) group. For a subgroup H of G, define X H = {x|hx = x for all h ∈ H} ⊂ X; X H is the H-fixed point subspace of X. Topologize the set of functions G/H −→ X as the product of copies of X indexed on the elements of G/H, and give the set of G-maps G/H −→ X the subspace topology. 5. Show that the space of G-maps G/H −→ X is naturally homeomorphic to X H .

We generally write B ∪g X for the pushout of a given cofibration i : A −→ X and a map g : A −→ B. 43 44 COFIBRATIONS Lemma. If i : A −→ X is a cofibration and g : A −→ B is any map, then the induced map B −→ B ∪g X is a cofibration. Proof. Notice that (B ∪g X)×I ∼ = (B ×I)∪g×id (X ×I) and consider a typical test diagram for the HEP. The proof is a formal chase of the following diagram: A qq qq g qq qq qq 5 i0 G A×I o o o g×idoo ooo o o ow G B×I r r h rr rrr r r rx r pushout pushout i×id i Yw Yy1 eu u w u ˜ f ww h u 1 w uu ww  1  ww † † G (B ∪g X) × I B ∪g X † † gxx wY † † ww † † xxxxx w † † ww ¯ † †xxxx  h  www † G X × I.

However, our interest here is in discrete groups G, for which the continuity condition just means that action by each element of G is a homeomorphism. The functoriality on O(G) of our construction of general covers will be immediate from the following observation. Lemma. Let X be a G-space. 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.

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