(i) True or False: If a path-connected space X is contractible (i.e. the identity map id : X X. id(r) = x, is homotopic to a constant map c: X→ X, c(a) = x0), then X is simply connected (i.e. T1(X, xo) is trivial). The converse is not necessarily true.
(i) True or False: If a path-connected space X is contractible (i.e. the identity map id : X X. id(r) = x, is homotopic to a constant map c: X→ X, c(a) = x0), then X is simply connected (i.e. T1(X, xo) is trivial). The converse is not necessarily true.
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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Topology
Please don't reject saying not clear image.
![(i) True or False: If a path-connected space X is contractible (i.e. the identity map
id: X X. id(z) = r, is homotopic to a constant map c: X→ X, c(x) = 10), then
X is simply connected (i.e. 7(X, to) is trivial). The converse is not necessarily true.
(ii) Assume X is path connected and 7(X, 2o) is abelian (ie. [f] lg = [g] * [f] for all
F], Lg] E T1(X, Lo)). Show that for any yo € Y and any two paths aq 02 connecting
1o to yo the homomorphisms Pı, P2 : 71(X.26) → TI(X, 30) with ở (/]) = [a,'fa]
and 2(f]) = [a, fa) are the same homomorphism.
E3
(iii) Let p: X→X be a covering map and assume both X and X are path connected. Say
a0, 1 X is a loop based at ro E X and assume go, o ep(ro}, jo 2o. If the
unique lift a: (0, 1] X with a(0) = is such that a(1)= , then X is not simply
connected.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F265f0d2b-afd7-4e04-8900-233676489eb3%2F6d8881c7-4fc0-4fa5-8c93-4d0c74c1fe1a%2F26fmx4b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(i) True or False: If a path-connected space X is contractible (i.e. the identity map
id: X X. id(z) = r, is homotopic to a constant map c: X→ X, c(x) = 10), then
X is simply connected (i.e. 7(X, to) is trivial). The converse is not necessarily true.
(ii) Assume X is path connected and 7(X, 2o) is abelian (ie. [f] lg = [g] * [f] for all
F], Lg] E T1(X, Lo)). Show that for any yo € Y and any two paths aq 02 connecting
1o to yo the homomorphisms Pı, P2 : 71(X.26) → TI(X, 30) with ở (/]) = [a,'fa]
and 2(f]) = [a, fa) are the same homomorphism.
E3
(iii) Let p: X→X be a covering map and assume both X and X are path connected. Say
a0, 1 X is a loop based at ro E X and assume go, o ep(ro}, jo 2o. If the
unique lift a: (0, 1] X with a(0) = is such that a(1)= , then X is not simply
connected.
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