answer:To determine the fundamental group of R3 with a line removed, we can use a few different approaches, but one of the most insightful is to consider a deformation retract. Imagine R3 with a line, say the z-axis, removed. This space is homotopy equivalent to a space that is easier to analyze. We can deformation retract this space onto a circle, say in the xy-plane, centered at the origin. The idea is to radially project all points in R3 minus the z-axis onto this circle, effectively shrinking all radial lines from the z-axis onto the circle. This deformation retract preserves the topological properties we're interested in. Now, we know that the fundamental group of a circle is the integers, denoted by Z. Since the deformation retract preserves the fundamental group, we conclude that the fundamental group of R3 with a line removed is also Z. This result can also be understood in terms of loops and the concept of going around the removed line. Any loop in R3 with a line removed can be deformed into a loop that either does not encircle the removed line, which would be homotopic to a point (and thus trivial in the fundamental group), or one that does encircle it. In the latter case, how many times the loop wraps around the line matters, which corresponds to the integer values in Z. Thus, removing a line from R3 creates a non-trivial topological feature that is reflected in its fundamental group, illustrating how even in a seemingly simple space, interesting topological properties can emerge when parts are removed.

answer:When we remove two lines from R3, the situation becomes slightly more complicated than removing just one line. To determine the fundamental group of this space, we need to consider how loops can now interact with both removed lines. Assume the two lines are skew (i.e., they do not intersect and are not parallel). We can deformation retract the space R3 with two lines removed onto a space that is easier to analyze, much like we did in the case of one removed line. This time, however, the deformation retract will give us a bouquet of two circles (two circles joined at a single point). Each circle corresponds to one of the removed lines. The fundamental group of a bouquet of two circles is the free group on two generators, denoted by F2. This is because loops can now wrap around the first line, the second line, or both, in any combination, but these combinations cannot be simplified or unwound as they could if the lines were not present. The free group F2 can be thought of as consisting of words formed from two letters (the generators, corresponding to loops around each line), their inverses (corresponding to loops around the lines in the opposite direction), and the identity element. This captures all possible combinations of encircling the two removed lines. Thus, the fundamental group of R3 with two lines removed is F2, a more complex group than the integers Z we obtained with one line removed, reflecting the richer topological structure that arises from the presence of two distinct obstacles for loops to encircle.

answer:Removing two parallel lines from R3 results in a space that is topologically different from the case of two skew lines. To find the fundamental group, we can again consider a deformation retract, but this time, the retract will reveal a different structure. Imagine the two parallel lines being the z-axis and a line parallel to it, say at x = 1. We can deformation retract R3 with these two lines removed onto a space consisting of two circles joined by a line segment (not just at a single point as in the case of skew lines). However, this space can be further retracted onto a single circle. To see this, visualize a plane perpendicular to the two lines. Within this plane, the deformation retract will initially give two circles, but since these circles are connected by a line segment (lying in the plane), we can further retract the entire space onto just one of these circles. Essentially, loops around the two lines can be deformed into each other, showing that encircling one line is not fundamentally different from encircling the other. Since the space retracts onto a single circle, the fundamental group of R3 with two parallel lines removed is the integers, Z. This is the same fundamental group we obtained for R3 with one line removed, indicating that removing a second line parallel to the first does not introduce any new, fundamentally distinct holes or obstacles for loops to encircle.

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