A Concise Course in Algebraic Topology (Chicago Lectures in by J. P. May
By J. P. May
J. Peter May's technique displays the big inner advancements inside of algebraic topology during the last numerous many years, such a lot of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical shows of assorted themes the place acceptable. so much chapters finish with difficulties that additional discover and refine the suggestions 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 publication concludes with an inventory of recommended readings for these drawn to delving additional into the field.
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Let Cov(B) denote the category of coverings of the space B; when B is understood, we write Cov(E, E ′ ) for the set of maps E −→ E ′ of coverings of B. 7. THE CLASSIFICATION OF COVERINGS OF SPACES 29 Lemma. A map g : E −→ E ′ of coverings is itself a covering. Proof. The map g is surjective by the algebraic analogue. The fundamental neighborhoods for g are the components of the inverse images in E ′ of the neighborhoods of B which are fundamental for both p and p′ . The following remarkable theorem is an immediate consequence of the fundamental theorem of covering space theory.
If i : Y −→ M f is the inclusion, then r ◦ i = id and id ≃ i ◦ r. In fact, we can define a deformation h : M f × I −→ M f of M f onto i(Y ) by setting h(y, t) = y and h((x, s), t) = (x, (1 − t)s). It is not hard to check directly that j : X −→ M f satisfies the HEP, and this will also follow from the general criterion for a map to be a cofibration to which we turn next. 4. A criterion for a map to be a cofibration We want a criterion that allows us to recognize cofibrations when we see them. We shall often consider pairs (X, A) consisting of a space X and a subspace A.
Then k −→ t(k) specifies an isomorphism between K and the group Aut(H). Let X and Y be connected, locally path connected, and Hausdorff. A map f : X −→ Y is said to be a local homeomorphism if every point of X has an open neighborhood that maps homeomorphically onto an open set in Y . 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.