Questions: Simply Connected Spaces

Question 1

Which of the following spaces is NOT simply connected?

• A. The plane ℝ²
• B. The 2-sphere S²
• C. The circle S¹
• D. Any contractible space

Answer: (C)

S¹ is not simply connected because its fundamental group π₁(S¹) = ℤ — a loop that winds around the circle once cannot be contracted to a point without leaving S¹. The plane ℝ² and any contractible space are simply connected (π₁ = trivial). S² is also simply connected: any loop on a sphere can be continuously pulled to the north pole, so π₁(S²) = {e}. The winding-number obstruction is the precise 1-dimensional 'hole' that simple connectivity rules out.

Question 2

You remove a single point from ℝ². What happens to the simple connectivity of the resulting space?

• A. It remains simply connected — removing one point doesn't create a loop
• B. It loses simple connectivity — the puncture creates a non-contractible loop
• C. It becomes simply connected with a different basepoint far from the removed point
• D. Simple connectivity is undefined for spaces with missing points

Answer: (B)

ℝ² minus a point deformation retracts onto a circle surrounding that point, so it has fundamental group π₁ = ℤ — the same as S¹. A loop encircling the puncture cannot be contracted to a point without passing through the hole. Simple connectivity fails everywhere, not just near the removed point, because the group is a topological invariant of the whole space. This is why complex analysis requires simply connected domains: if the domain has a puncture, contour integrals around it may be non-zero.

Question 3

A simply connected space should have trivial homotopy groups πₙ for most n ≥ 1.

• T. True
• F. False

Answer: False

Simple connectivity only requires π₁ = {e} (trivial fundamental group). Higher homotopy groups can be non-trivial. The 2-sphere S² is simply connected — every loop on S² can be contracted to a point — yet π₂(S²) = ℤ, reflecting the existence of non-contractible 2-spheres mapped into S². Simple connectivity rules out 1-dimensional holes, not higher-dimensional ones. This is why it appears as a hypothesis in theorems about 1-dimensional objects (paths, loops, line integrals) but not as a blanket guarantee of topological triviality.

Question 4

The circle S¹ is not simply connected because there exist loops based at any point that cannot be continuously deformed to the constant loop while staying in S¹.

• T. True
• F. False

Answer: True

This is precisely the definition of failing simple connectivity. A loop that winds around S¹ once represents the generator of π₁(S¹) = ℤ, and no continuous deformation within S¹ can shrink it to a point. Any deformation that tries to collapse it would have to pass through the 'inside' of the circle, which is not part of S¹. The integer winding number is a topological invariant — it cannot change under continuous deformation — so the loop and the constant loop are in different homotopy classes.

Question 5

Why does Cauchy's integral theorem in complex analysis require the domain to be simply connected, and what goes wrong if the domain has a 'hole'?

Model answer: Cauchy's theorem states that the integral of a holomorphic function around any closed loop in a simply connected domain is zero. Simple connectivity ensures that any loop in the domain bounds a disk also contained in the domain — the loop can be contracted, and the integral over each infinitesimal piece cancels. If the domain has a hole (e.g., ℂ minus a point), a loop encircling the hole cannot be contracted; its integral may be non-zero (it equals 2πi times the residue at the missing point). The hole creates a non-contractible loop, and the path-dependence of the integral is precisely the topological obstruction that simple connectivity eliminates.

This is the canonical application of simple connectivity in analysis. The 1/z function on ℂ {0} integrates to 2πi around the origin — not zero — because the puncture prevents the loop from being contracted. In a simply connected domain, no such obstruction exists, and holomorphic functions automatically have antiderivatives. The algebraic topology of the domain (π₁) directly controls the analytic behavior of functions on it.