Van Kampen's theorem says π₁(X) ≅ π₁(U) *_{π₁(U∩V)} π₁(V). If π₁(U∩V) is trivial (the intersection is simply connected), what is π₁(X)?
Aπ₁(U) × π₁(V) — the direct product, with generators from both pieces commuting
Bπ₁(U) * π₁(V) — the free product, with no relations between the two sets of generators
Cπ₁(U) ⊕ π₁(V) — the direct sum
DThe trivial group, because a simply connected intersection means the pieces cannot have independent loops
The amalgamated free product imposes relations only through the image of π₁(U∩V). When π₁(U∩V) is trivial, there are no relations to impose — loops from U and loops from V are completely independent, generating a free product. The free product ℤ * ℤ (for the wedge of two circles) has generators that do NOT commute in general. Option A is the key misconception: trivial intersection gives the free product (no forced commutativity), not the direct product.
Question 2 Multiple Choice
In computing π₁(T²) via van Kampen's theorem, the relation aba⁻¹b⁻¹ = e arises from which part of the construction?
AThe definition of the fundamental group of a circle, which is abelian
BThe boundary loop of the square — it is contractible in V (the disk), so it must equal the identity in the amalgamated product
CThe fact that the torus is a product space S¹ × S¹, which forces commutativity
DThe Euler characteristic of the torus, which equals zero
U is the torus minus a small open disk (deformation-retracting to the boundary square representing the loop aba⁻¹b⁻¹), and V is a small open disk (simply connected). Their intersection U ∩ V is an annulus whose core circle represents aba⁻¹b⁻¹ when viewed in U, but this circle bounds the disk V, making it contractible (= identity) there. Van Kampen imposes that this loop equals the identity, giving aba⁻¹b⁻¹ = e, hence ab = ba. The commutativity comes entirely from the intersection relation — not from the torus being a product.
Question 3 True / False
Van Kampen's theorem applies even when the intersection U ∩ V is not path-connected.
TTrue
FFalse
Answer: False
The standard form of van Kampen's theorem requires U, V, and U ∩ V all to be open and path-connected. Path-connectivity of the intersection is crucial: it ensures a single basepoint can be used consistently across both U and V, and that the inclusion maps π₁(U ∩ V) → π₁(U) and π₁(U ∩ V) → π₁(V) are well-defined group homomorphisms. When U ∩ V is not path-connected, a more general formulation involving groupoids is required.
Question 4 True / False
π₁(S¹ ∨ S¹) is the free group on two generators (not ℤ × ℤ) because there is no relation in the construction that forces the two generating loops to commute.
TTrue
FFalse
Answer: True
By van Kampen's theorem with simply connected intersection, π₁(S¹ ∨ S¹) = π₁(U) * π₁(V) = ℤ * ℤ — the free product where generators a and b have no relation between them, so ab ≠ ba in general. The torus has the same generators but a different intersection contribution: the boundary relation aba⁻¹b⁻¹ = e forces commutativity, yielding ℤ × ℤ. Comparing these two cases shows that the topology of the intersection entirely determines whether generators commute.
Question 5 Short Answer
Explain the role of π₁(U ∩ V) in van Kampen's theorem. Why does the intersection determine whether generators of π₁(U) and π₁(V) commute in π₁(X)?
Think about your answer, then reveal below.
Model answer: The intersection U ∩ V is the gluing zone where U and V overlap. Any loop in U ∩ V can be viewed as a loop in U (via U ∩ V ↪ U) or as a loop in V (via U ∩ V ↪ V). Van Kampen's theorem says these two viewpoints must agree in the amalgamated product: the image of a loop γ ∈ π₁(U ∩ V) in π₁(U) must equal its image in π₁(V). This is the only source of relations. If U ∩ V is simply connected, no loops exist there to impose relations, so generators are fully independent (free product). If the intersection contains a loop that 'wraps around' generators from both sides, it forces a relation between them — as in the torus, where the boundary loop being contractible in V forces ab = ba.
Intuitively: you can only relate generators from different pieces of X if there is topological 'room' in the intersection for loops that see both sides. A simply connected intersection is too small to relate anything; a richer intersection provides the wire connecting the two pieces and determines what the combined group looks like.