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(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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.
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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