Question

Let (X, x 0) and (Y, y0) be given and let (X × Y, (x0, y0)) be their product. (a) Show that for any (K, k0) that [K, X × Y ] = [K, X ] × [K, Y ]. In particular pi1(X × Y ) = π1(X ) × pi1(Y ) (b) This implies that if a is a loop in X and b a loop in Y and we let A(t) = (a(t), yo) and B(t) = (xo, b(t)) then A and B commute in pi1(X ×Y ). Show this by finding an explicit homotopy between AB and BA

Name: let x x 0 and y y0 be given and let x y x0 y0 be their product a show that for any k k0 that k x y k x k y in particular pi1x y 1x pi1y b this implies that if a is a loop in x and b a loop i 73355
Uploaded: 2024-04-29T09:10:57-08:00
Duration: 6 min 31 s
Channel: Adi S
Description: let x x 0 and y y0 be given and let x y x0 y0 be their product a show that for any k k0 that k x y k x k y in particular pi1x y 1x pi1y b this implies that if a is a loop in x and b a loop i 73355

          Let (X, x 0) and (Y, y0) be given and let (X × Y, (x0, y0)) be their product.
(a) Show that for any (K, k0) that [K, X × Y ] = [K, X ] × [K, Y ]. In particular pi1(X × Y ) = π1(X ) × pi1(Y )
(b) This implies that if a is a loop in X and b a loop in Y and we let A(t) = (a(t), yo) and B(t) = (xo, b(t)) then A and B commute in pi1(X ×Y ). Show this by finding an explicit homotopy between AB and BA

Added by Roberto R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

04/29/2024

Step 1

The fundamental group \( \pi_1 \) of a space at a base point captures the loops based at that point up to homotopy. Show more…

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Let (X, x 0) and (Y, y0) be given and let (X × Y, (x0, y0)) be their product. (a) Show that for any (K, k0) that [K, X × Y ] = [K, X ] × [K, Y ]. In particular pi1(X × Y ) = π1(X ) × pi1(Y ) (b) This implies that if a is a loop in X and b a loop in Y and we let A(t) = (a(t), yo) and B(t) = (xo, b(t)) then A and B commute in pi1(X ×Y ). Show this by finding an explicit homotopy between AB and BA