MATH5665: Algebraic Topology- Course notes
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چکیده
These are the lecture notes for an Honours course in algebraic topology. They are based on standard texts, primarily Munkres’s “Elements of algebraic topology” and to a lesser extent, Spanier’s “Algebraic topology”. 1 What’s algebraic topology about? Aim lecture: We preview this course motivating it historically. Recall that a continuous map f : X −→ Y of topological spaces is a homeomorphism if it is bijective and f−1 is also continuous. In this case we say X and Y are homeomorphic and write X ' Y . Major Q in topology is how to determine if two spaces X,Y are homeomorphic or not. A depends on whether they are homeomorphic or not. If they are, one usually guesses the homeomorphism. If not, one needs to be a bit more sophisticated. One usually uses invariants to distinguish them. Historically, the first example is below. Euler Characteristic Notation 1.1 We denote the n-dim unit ball by B := {~v ⊂ R ||~v| ≤ 1} and the n-dim unit sphere by S := {~v ⊂ R ||~v| = 1} Consider a regular polyhedron K ⊂ R or more generally, any polyhedron homeomorphic to the unit sphere S. Let V = no. vertices, E = no. edges and F = no. faces. Theorem 1.2 (Euler) V − E + F = 2. E.g. Check the cube, tetrahedron etc. The quantity e(K) := V − E + F is called the Euler number of K. Quotient spaces Let X = topological space and ∼ be an equivalence relation on the underlying set so there is a set map π : X −→ X/ ∼: x 7→ [x] where X/ ∼ is the set of equivalence classes [x]. We put the strongest topology on X/ ∼ so that π is continuous, viz, U ⊂ X/ ∼ is open iff π−1(U) is open. The resulting topological space is called the quotient space. E.g. Let I = [0, 1] ⊂ R and X = I × I. We put the weakest equivalence relation on X s.t. (0, x) ∼ (1, x), (x, 0) ∼ (x, 1) for x ∈ I. We sometimes sum up this info in the following picture:
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تاریخ انتشار 2015