A student claims: 'A disk D² and a single point {p} cannot be homotopy equivalent, because D² has infinitely many points while {p} has only one — you cannot continuously deform one into the other.' Which response is correct?
AThe student is right: homotopy equivalence requires the spaces to be homeomorphic, and D² and {p} are not homeomorphic.
BThe student is wrong: D² and {p} are homotopy equivalent because the constant map D² → {p} and the inclusion {p} → D² compose (in both orders) to maps homotopic to the respective identity maps.
CThe student is right that homotopy equivalence fails, but the reason is that D² is 2-dimensional while {p} is 0-dimensional.
DThe student is partially right: D² and {p} are path-connected but not homotopy equivalent.
Homotopy equivalence is strictly weaker than homeomorphism. Two spaces are homotopy equivalent if there are maps f: X → Y and g: Y → X with g∘f ≃ id_X and f∘g ≃ id_Y. For D² and {p}: the constant map c: D² → {p} and inclusion ι: {p} → D² satisfy ι∘c(x) = p for all x, and the straight-line homotopy H(x,t) = (1−t)x contracts D² to its center — a homotopy from ι∘c to id_{D²}. The other composition is automatically the identity. So D² ≃ {p}, even though they are very different as spaces.
Question 2 Multiple Choice
What is the defining property of a homotopy H between maps f, g: X → Y?
AH is a bijective continuous map from X to Y that continuously deforms f into g.
BH is a continuous map H: X × [0,1] → Y with H(x,0) = f(x) and H(x,1) = g(x) for all x ∈ X.
CH is any map satisfying H(x,0) = f(x) and H(x,1) = g(x), with no continuity requirement.
DH is a homeomorphism between X × [0,1] and a subspace of Y.
A homotopy is a continuous map on the product space X × [0,1], with the unit interval playing the role of a 'time' parameter. At t = 0 you get f; at t = 1 you get g; continuity of H on the whole product ensures the deformation has no jumps or tears. The bijection and homeomorphism conditions (options A and D) are far too strong — homotopies need not be bijective. Dropping continuity (option C) loses the essential content.
Question 3 True / False
Homotopy equivalence is weaker than homeomorphism: every pair of homeomorphic spaces is homotopy equivalent, but not every homotopy equivalent pair is homeomorphic.
TTrue
FFalse
Answer: True
A homeomorphism f: X → Y with inverse g satisfies g∘f = id_X and f∘g = id_Y as equalities, which are in particular homotopies (the constant homotopy H(x,t) = g(f(x)) = x works). So homeomorphism implies homotopy equivalence. The converse fails: D² and a point are homotopy equivalent but not homeomorphic. Homotopy equivalence allows collapsing and re-expanding in ways that homeomorphism does not.
Question 4 True / False
If two continuous maps f and g from X to Y are homotopic, then for each fixed point x ∈ X, the path t ↦ H(x,t) is a loop in Y based at f(x).
TTrue
FFalse
Answer: False
The path t ↦ H(x,t) goes from H(x,0) = f(x) to H(x,1) = g(x). It is a loop only if f(x) = g(x) for that particular x. In general, homotopic maps can send each point to completely different images, and the paths traced by individual points need not be closed. Loops in Y based at a point are the input to the fundamental group, which is a different and more specific construction.
Question 5 Short Answer
What is the difference between two maps f and g being homotopic and two spaces X and Y being homotopy equivalent?
Think about your answer, then reveal below.
Model answer: Map homotopy (f ≃ g) is a relation between two maps with the same domain and codomain: there exists a continuous deformation H: X × [0,1] → Y interpolating between them. Space homotopy equivalence (X ≃ Y) is a relation between two spaces: there exist maps f: X → Y and g: Y → X such that g∘f ≃ id_X and f∘g ≃ id_Y. The round-trip conditions mean each space can be mapped into the other and back with only a homotopic-to-identity distortion. Every homeomorphism gives a homotopy equivalence, but a disk and a point are homotopy equivalent without being homeomorphic.
The distinction matters because the fundamental group and other homotopy invariants are invariants of homotopy equivalence classes of spaces — not of individual maps. Homotopy of maps is the tool used to *define* homotopy equivalence of spaces.