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# Elementary Topology MA 57100

Purdue

GPA 3.97

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This 7 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 57100 at Purdue University taught by Staff in Fall. Since its upload, it has received 53 views. For similar materials see /class/208137/ma-57100-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

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Let 1900 90 a Give the definition of the standard map 9 W1B0 gt pquot0 constructed in Munkres you do NOT have to prove that this is well defined b Suppose that x and 3 are two elements of W1 B 90 with 9a 93 Prove that there is an element 7 of W1E 0 With 3 pt r Let p R gt S be the usual covering map specifically pt cos 2m sin 2W0 Let 0 6 S1 be the point 1 0 Recall that the standard map 9 W1Sl0 gt Z is defined by f1 Where f is a lifting of f With f0 0 14 points Prove that 9 is 1 1 b 14 points Prove that 9 is a group homomorphism Let p E gt B be a covering map Let 0 6 E and let I90 pa0 Prove that 19 W1Ea0 gt mBb0 is 11 Let p Y gt X be a covering map let 3 6 Y and let x Let I be a loop beginning and ending at x and let a be the corresponding element of W1 X 93 Let 3 be the unique lifting of I to a path starting at y Prove that if a 6 pm1Y y then 3 ends at 3 Let I be the unit interval and let Y be a path connected space Prove that any two maps from I to Y are homotopic Let X be a topological space and f 0 1 gt X any continuous function Define f by f t f 1 t Prove that f gtlt f is path homotopic to the constant path at f 0 Let X and Y be topological spaces and let f X gt Y be a continuous function Let 4 6 X and let go points Give the definition of the function m X 330 gt m Y yo including the proof that it is well defined b 10 points Prove that if f is a covering map then is one to one 390 3M 30 3311x331 Ixogmmtgap 12 8 IS 3mg suomugap aq mag QAOJd 39zg u ug gm aq aq 0 36 39zg ug I xxlzx x amp aq aq IS 331 39 39X lt 15 sdmn snonugum paszzq 30 sasszzp Kdmomoq pGSRQ am pm 058 qu Imammq a3mpuodsaxxm 1 p sg mam 3mg aond 3915 30 mmd asuq aq aq 0 1 071 mi pm 02 mmdaszzq mm aazzds p aq X 331 391 up 103 02 1 0 711H 3g pawoq aq 03 p328 sg Z lt 1 X M H Kdmomoq p pm 02 0mf 3g pawoq aq 03 p328 sg Z lt M f hm p 02 mmd asuq mm amds p sg Z pm 0m mmd asuq mm aazzds p sg M 31 39uogggugaq 39 39dmn mmsum p 03 agdmomoq sg IS 0 25 mag hm snonunum KJGAG 3mg aond 39 39sammmw dnm aq mm 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Sugqmtm 3st 5mm my 3931 gastxf a1mmxaw3unm p ng mm 11 pm a1hmzxaw3unm p ng xo aAmd 39 81 017 mm lt Ox X 11 f uaq 1 81 f 11 11am mmxmms Sumoutg mp SI 39027 03 02 mg ngm Hanging snonugum p aq A lt X f 331 pm A 9 017 X 9 0x 331 samds pag qodm aq A pm X 331 39 x 39I L v lt and U V U D V are path connected Lot 171 U D V gt U 172 U D V gt V jl U gt X and 2 V gt X be the inclusion maps Suppose that Mt mU Vim gt W1Ux0 is onto Prove using the Seifei t van Kampen theorem that 726 W1Vx0 gt W1Xx0 is onto

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