Q1. Let X ⊂ R3 be the union of n lines through the origin. Compute π1(R3\X).
Q2. Let X be the space obtained by taking two copies of a torus S1 × S1 and glueing a circle S1 × {x0} in one torus with the same circle in the other torus. Compute π1(X).
Let X1 and X2 be surfaces. The connected sum of X1 and X2 is the surface X1#X2 obtained by deleting the interior of a closed discs D1 ⊂ X1 and D2 ⊂ X2 and identifying the resulting boundary circles ∂D1 and ∂D2 via some homeomprphism between them. (For example, if X is a surface and T = S1 × S1, then X#T is X with a handle attached, X#RP2 is X with a cross-cap attached, and X#S2 = X.)
Q3. Compute π1(Mg) where Mg is the genus g orientable surface in two ways:
(a) First by considering Mg = Mg-1#T, where T = S1 × S1.
(b) Second by considering Mg as the quotient of a regular 4g-gon by identifying pairs of edges in some appropriate way.
Q4. Compute π1(K) where K is the Klein bottle in four ways:
(a) First by considering K to be the quotient of S1 × I obtained by identifying (z, 0) with (-z, 1) for all z ∈ S1.
(b) Second by considering K = T #RP2.
(c) Third by considering K = RP2#RP2.
(d) Fourth by considering K as the quotient of a square with edges identified in the usual way.