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Def: Let p : E → B be a map. If f : X → B is a map, a lifting of f is a map
f : X → E such that p ◦
f = f Diagrams: E p
X f
f B (E,e0) p
(I,0) f
f
(B,b0) Lemma: (Path Lifting) Let p : E → B be a covering map. Let p(e0) = b0. Any path f : [0,1] → B beginning at b0 has a unique lifting to a path
f in E beginning at e0. Mth 632 – Winter 2009 Path and Homotopy Lifting 1/5 Homotopy Lifting Lemma: (Homotopy Lifting) Let p : E → B be a covering map. Let p(e0) = b0. Any homotopy H : [0,1]×[0,1] → B with H(0) = b0 has a unique lifting to a homotopy H
in E with H(0) = e0. If H is a path homotopy, so is H
. (E,e0) p
(I ×I,0) H H
(B,b0) Mth 632 – Winter 2009 Path and Homotopy Lifting 2/5 Lifting Correspondence Thm: Let p : E → B be a covering map. Let p(e0) = b0. paths f and g from b0 to b1 are path homotopic iff
f and
g, the lifts of f and g beginning at e0, end at the same point and are path homotopic. Def: Let p : E → B be a covering map. Let b0 ∈ B. Choose an e0 so that p(e0) = b0. Given [f] ∈ π1(B,b0), let
f be the lifting of f to a path in E beginning at e0. Let φe0(f) be
f(1). φe0 : π1(B,b0) → p−1(b0) is called the lifting correspondence derived from p. Mth 632 – Winter 2009 Path and Homotopy Lifting 3/5 Fundamental G |
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Posted On: 03/05/2019 10:07AM | View jiggaloon's Profile | # |