Problem 7 The Möbius strip is defined to be the quotient space M = 0,1²/ ~, where (0. y) (1, 1 – 4), y € 0.1 are the only non-trivial identifications. Is the boundary circle of the Möbius strip, i.e. the set = {[(7.v)| € M3 € 0, 1]| y € {0, 1}} a retract of M? If so, find à retraction r: M → ƏM. If not, prove why. Hint(s): First prove that #1(M) = Z by showing the circle through the center of M par- allel to the unidentified sides (i.e. the image of the line [0, 1] × {1/2} under the quotient map) is a deformation retract of M. Notice that the boundary of the Möbius strip is homeomorphic to a circle. Consider the loop w: [0, 1]→ ƏM which goes around the boundary exactly once and convince yourself that wi E 7;(M) this not in the homotopy class of the loop that goes around M once. With this in mind. determine explicitly the map i, : 7i(ðM)→ M induced by inclusion i : ÔM → M (you might as well regard #1(ðM) as Z), Now assume there is a retraction r : M → ƏM. Consider the induced map (r oi), and determine what must be the value of r.(wi) to derive a contradiction.
Problem 7 The Möbius strip is defined to be the quotient space M = 0,1²/ ~, where (0. y) (1, 1 – 4), y € 0.1 are the only non-trivial identifications. Is the boundary circle of the Möbius strip, i.e. the set = {[(7.v)| € M3 € 0, 1]| y € {0, 1}} a retract of M? If so, find à retraction r: M → ƏM. If not, prove why. Hint(s): First prove that #1(M) = Z by showing the circle through the center of M par- allel to the unidentified sides (i.e. the image of the line [0, 1] × {1/2} under the quotient map) is a deformation retract of M. Notice that the boundary of the Möbius strip is homeomorphic to a circle. Consider the loop w: [0, 1]→ ƏM which goes around the boundary exactly once and convince yourself that wi E 7;(M) this not in the homotopy class of the loop that goes around M once. With this in mind. determine explicitly the map i, : 7i(ðM)→ M induced by inclusion i : ÔM → M (you might as well regard #1(ðM) as Z), Now assume there is a retraction r : M → ƏM. Consider the induced map (r oi), and determine what must be the value of r.(wi) to derive a contradiction.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Problem 7
The Möbius strip is defined to be the quotient space M= 0,1²/ ~, where (0. y).
(1,1-y). y € (0.1 are the only non-trivial identifications. Is the boundary circle of the
Möbius strip, ie. the set
OM = {(7.y)| E M€ 0,1| y € {0, 1}}
a retract of M? If so, find à retraction r:M →@M. If not, prove why.
Hint(s): First prove that (M) = Z by showing the circle through the center of M par-
allel to the unidentified sides (i.e. the image of the line (0, 1 ×{1/2} under the quotient map)
is a deformation retract of M. Notice that the boundary of the Möbius strip is homeomorphic,
to a circle. Consider the loop wi : 0,1]→ ƏM which goes around the boundary exactly
once and convince yourself that wiE 7;(M) this not in the homotopy class of the loop that
goes around M once. With this in mind, determine explicitly the map i, : TilðM)→M
induced by inclusion i: OM → M (you might as well regard T1(OM) as Z). Now assume
there is a retraction r: M → aM. Consider theinduced map (r oi), and determine what
must be the alue of r.(wD to derive a contradiction.
DII FS
End F10
PgUp Fw
Prt Scn F8
Home F9
F6
F7
%
&
5
6
7
8.
9.
Y
U](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F265f0d2b-afd7-4e04-8900-233676489eb3%2F69d3c764-1b29-4c9f-8a1a-3862d455f811%2Fr2ewsil_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 7
The Möbius strip is defined to be the quotient space M= 0,1²/ ~, where (0. y).
(1,1-y). y € (0.1 are the only non-trivial identifications. Is the boundary circle of the
Möbius strip, ie. the set
OM = {(7.y)| E M€ 0,1| y € {0, 1}}
a retract of M? If so, find à retraction r:M →@M. If not, prove why.
Hint(s): First prove that (M) = Z by showing the circle through the center of M par-
allel to the unidentified sides (i.e. the image of the line (0, 1 ×{1/2} under the quotient map)
is a deformation retract of M. Notice that the boundary of the Möbius strip is homeomorphic,
to a circle. Consider the loop wi : 0,1]→ ƏM which goes around the boundary exactly
once and convince yourself that wiE 7;(M) this not in the homotopy class of the loop that
goes around M once. With this in mind, determine explicitly the map i, : TilðM)→M
induced by inclusion i: OM → M (you might as well regard T1(OM) as Z). Now assume
there is a retraction r: M → aM. Consider theinduced map (r oi), and determine what
must be the alue of r.(wD to derive a contradiction.
DII FS
End F10
PgUp Fw
Prt Scn F8
Home F9
F6
F7
%
&
5
6
7
8.
9.
Y
U
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