a compute the fundamental group of a topological space X obt

a)

compute the fundamental group of a topological space X obtained from S^2 by removing from it two distinc points p,q.

b) compute the fundamental group of the product of two circles with the removed diagonal.

c) compute the fundamental group of the product space S^1 X S^2

Problem 4 (a) Compute the fundamental group of a topological space X obtained from S by removing from it two distinct points p,q. (b) Compute the fundamental group of the product of two circles with the removed diagonal. In other words, find 1 (S1 xgl-A), where = { (z, y) E S1xSlp = y}. (c) Compute the fundamental group of the product space S1 S2.

Solution

To give a more geometric way of viewing this. Recall that S1×S1S1×S1 is homeomorphic to the torus T2T2 which we can view as a square [0,1]×[0,1][0,1]×[0,1] with opposite edges glued together in the usual way. Take this square and cut out the diagonal ={x,yx=y}={x,yx=y}. Now glue the left edge to the right edge to leave us with a parallelogram with the top and bottom edges glued together in the same orientation. Well, by a bijective linear transformation of the plane (a shear), we can homeomorphically map this parallelogram onto a square with one pair of opposite edges glued together. This is (one of the) usual model for the cylinder which is homotopy equivalent to the circle and so 1(T2)1(S1)Z1(T2)1(S1)Z.

1(S 1 × S 2 ) = 1(S 1 ) × 1(S 2 ) = Z × {e} = Z.