shortest-homotopic-paths - Computational Topology (Jeff...

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Computational Topology (Jeff Erickson) Shortest Homotopic Paths Oh! marvellous, O stupendous Necessity–by thy laws thou dost compel every effect to be the direct result of its cause, by the shortest path. These [indeed] are miracles. .. — Leonardo da Vinci, Codex Atlanticus (c. 1500) translated by Jean Paul Richter (1883) Those who cannot remember the past are condemned to repeat it. — George Santayana, Reason in Common Sense (1905) A straight line may be the shortest distance between two points, but it is by no means the most interesting. — The Doctor [Jon Pertwee], The Time Warrior (1973) 2 Shortest Homotopic Paths 2.1 A Few Definitions Let X be any topological space. A path in X is a continuous function from the unit interval [ 0,1 ] to X , and a cycle in X is a continuous function from the standard unit circle S 1 : = { ( x , y ) R 2 | x 2 + y 2 = 1 } to X . A loop is a path whose endpoints coincide; this common endpoint is called the basepoint of the loop. A path or cycle is simple if it is injective; a loop is simple if its restriction to [ 0,1 ) is injective. We refer to paths, cycles, and loops collectively as curves . The Jordan-Schönflies Theorem states that for any simple cycle γ in the plane, there is a homeo- morphism from the plane to itself whose restriction to S 1 is γ . Thus, the image of any simple cycle partitions the plane into two components, a bounded interior whose closure is homeomorphic to the disk B 2 , and an unbounded exterior . For any paths π and π 0 with π ( 1 ) = π 0 ( 0 ) , the concatenation π · π 0 is the path ( π · π 0 )( t ) : = ( π ( 2 t ) if t 1 / 2, π 0 ( 2 t - 1 ) if t 1 / 2. The reversal π of a path π is the path π ( t ) : = π ( 1 - t ) . A path homotopy between paths π and π 0 is a continuous function h : [ 0,1 ] × [ 0,1 ] X such that h ( 0, t ) = π ( t ) and h ( 1, t ) = π 0 ( t ) for all t , and h ( s ,0 ) = π ( 0 ) = π 0 ( 0 ) and h ( s ,1 ) = π ( 1 ) = π 0 ( 1 ) for all s [ 0,1 ] . (We will omit the word ‘path’ when it is clear from context.) For all t [ 0,1 ] , the function s 7→ h ( s , t ) is a path from π ( 0 ) to π ( 1 ) . Two paths π and π 0 are (path) homotopic if there is a path homotopy between them; we write π π 0 to denote that π and π 0 are homotopic. Tedious definition-chasing implies that is an equivalence relation. We refer to the equivalence classes as homotopy classes , and write [ π ] to denote the homotopy class of path π . A homotopy between two paths. A homotopy from a contractible loop to its basepoint. We call a loop is contractible if it is path-homotopic to the constant path mapping the entire interval [ 0,1 ] to the basepoint ( 0 ) . Two paths π and π 0 with the same endpoints are homotopic if and only if the loop π · π 0 is contractible. A connected topological space X is simply connected if every loop 1
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Computational Topology (Jeff Erickson) Shortest Homotopic Paths in X is contractible. For example, the plane and the sphere are both simply connected, but the annulus
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shortest-homotopic-paths - Computational Topology (Jeff...

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