Let x1 and x2 be surfaces the connected sum of x1 and x2 is

Question

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(M_g) where Mg is the genus g orientable surface in two ways:

(a) First by considering M_g = M_g-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.