Consider two paths in the punctured plane ℝ² \ {0}, both going from (-1, 0) to (1, 0): path α travels above the origin, and path β travels below. Are α and β homotopic rel endpoints?
AYes — both paths connect the same endpoints, so they can always be continuously deformed into each other
BNo — the hole at the origin is a topological obstruction preventing any continuous deformation from α to β while fixing endpoints
CYes — but only if we allow the endpoints to move slightly during the deformation
DNo — because the paths have different arc lengths, so no length-preserving deformation exists
In ℝ² (no hole), any two paths with the same endpoints are homotopic via the straight-line homotopy H(t, s) = (1−s)α(t) + sβ(t). But in ℝ² \ {0}, any deformation from α to β must at some point pass through the origin — which is removed from the space. No continuous map H: [0,1]×[0,1] → ℝ² \ {0} can achieve the deformation, so α and β belong to different homotopy classes. Length is irrelevant to homotopy; only the topology of the space matters.
Question 2 Multiple Choice
Why is the requirement that endpoints remain fixed throughout a path homotopy essential, rather than merely a technical convenience?
AIt is purely technical — it simplifies the proof that homotopy is an equivalence relation
BWithout fixing endpoints, any two paths in a path-connected space could be deformed into each other, erasing all topological information about the space
CIt ensures the homotopy is differentiable rather than merely continuous
DIt prevents the deformation from reversing the orientation of the path
If endpoints were allowed to move freely (free homotopy), then in any path-connected space you could slide one endpoint to meet the other and shrink the path to a point, making every path homotopic to every other. All topological information would be lost. Fixing the endpoints forces the homotopy to measure how paths 'go around' obstacles — holes, handles, etc. — between two fixed points. The distinction between free homotopy and homotopy rel endpoints is precisely what makes the fundamental group a meaningful invariant.
Question 3 True / False
In ℝ², any two paths with the same endpoints are homotopic rel endpoints.
TTrue
FFalse
Answer: True
ℝ² is convex: for any two paths α and β from p to q, the straight-line homotopy H(t, s) = (1−s)α(t) + sβ(t) is a valid homotopy. It is continuous, satisfies H(t, 0) = α(t) and H(t, 1) = β(t), and fixes both endpoints H(0, s) = p and H(1, s) = q for all s. No obstructions exist because the space has no holes.
Question 4 True / False
A loop based at a point p that winds once around a hole and a loop that winds twice around the same hole are homotopic rel endpoints in any space containing that hole.
TTrue
FFalse
Answer: False
These loops belong to different elements of the fundamental group π₁(X, p). In the punctured plane ℝ² \ {0}, the fundamental group is ℤ, where the integer counts the winding number. A loop winding once corresponds to 1 ∈ ℤ and a loop winding twice corresponds to 2 ∈ ℤ. Since 1 ≠ 2, no homotopy rel endpoints exists between them. You cannot continuously deform one into the other without passing through the removed point.
Question 5 Short Answer
What determines whether two paths between the same endpoints belong to the same homotopy class, and what does this have to do with the shape of the space?
Think about your answer, then reveal below.
Model answer: Two paths are in the same homotopy class if and only if one can be continuously deformed into the other with endpoints fixed. Whether this is possible depends on the topology of the space — specifically, whether the region between the paths contains any holes or other obstructions. In simply connected spaces (no holes), all paths between the same endpoints are homotopic. In spaces with holes, paths that wind differently around those holes belong to different homotopy classes. The set of homotopy classes of loops at a basepoint, under concatenation, forms the fundamental group — which encodes the space's hole structure.
The key insight is that homotopy classes count topological obstructions. Two paths in the same class can be deformed into each other by sliding through the space continuously; paths in different classes are separated by a hole or feature that no continuous deformation can cross. This is why the fundamental group is a topological invariant: homeomorphic spaces have isomorphic fundamental groups, because a homeomorphism carries homotopy classes to homotopy classes.