Homotopy

The definition

Let X and Y be topological spaces and f, g : X → Y continuous maps. A homotopy from f to g is a continuous map

H : X × [0, 1] → Y

with H(x, 0) = f(x) and H(x, 1) = g(x) for every x ∈ X. Think of t ∈ [0, 1] as time and H_t(x) := H(x, t) as a snapshot at time t: a continuously varying family of maps starting at f and ending at g. If such an H exists we write f ≃ g and say f and g are homotopic.

Homotopy is an equivalence relation. Reflexivity: H(x, t) = f(x) for all t works. Symmetry: replace t by 1 − t. Transitivity: if H₁ goes f to g and H₂ goes g to h, then H(x, t) = H₁(x, 2t) for t ∈ [0, ½] and H₂(x, 2t − 1) for t ∈ [½, 1] is continuous by the pasting lemma. Equivalence classes of maps are called homotopy classes.

Paths, loops, and π₁

The most important special case is when X is the unit interval. A path from a to b in Y is a continuous γ : [0, 1] → Y with γ(0) = a and γ(1) = b. A loop at x₀ is a path with γ(0) = γ(1) = x₀.

Two loops γ, δ at x₀ are homotopic relative to {0, 1} if there is H : [0, 1] × [0, 1] → Y interpolating between them while keeping endpoints fixed: H(0, t) = H(1, t) = x₀ for all t. This refined relation lets us define the fundamental group π₁(Y, x₀): equivalence classes of loops under endpoint-fixing homotopy, with concatenation as the group operation. Identity is the constant loop; inverses are reversals. π₁(Y, x₀) was introduced by Henri Poincaré in his 1895 paper Analysis Situs — the founding moment of algebraic topology.

Homotopy vs neighbouring notions

Relation	Maps required	Constraint	Example: same?	Coarseness
Homotopy (f ≃ g)	Continuous family H_t	None on intermediate maps	Disc ≃ point (yes)	Coarsest of these notions
Homotopy equivalence (X ≃ Y)	f : X → Y, g : Y → X with f∘g ≃ id, g∘f ≃ id	Inverse up to homotopy	Möbius strip ≃ circle (yes)	Captures "essential shape"
Isotopy	Family H_t of homeomorphisms	Each H_t is a homeomorphism	Trefoil ≃ unknot? (no — different knots)	Stricter than homotopy
Ambient isotopy	Isotopy of ambient space carrying one map to another	Whole ambient deforms continuously	Used to define knot equivalence	Stricter still
Homeomorphism	Bijective continuous with continuous inverse	Single map, no "time"	S¹ ≃ ℝ? (no — non-homeomorphic)	Standard topological equivalence
Diffeomorphism	Smooth bijection with smooth inverse	Manifolds; smooth structure preserved	S⁷ has 28 inequivalent smooth structures (Milnor)	Finest natural notion for smooth manifolds

Worked example: paths on a plane and on an annulus

On ℝ², take two paths from (0, 0) to (1, 0): the straight line γ(s) = (s, 0) and the half-circle δ(s) = ((1 − cos πs)/2, sin πs / 2). Are they homotopic?

Yes. The straight-line homotopy

H(s, t) = (1 − t) γ(s) + t δ(s)

is continuous, with H(s, 0) = γ(s), H(s, 1) = δ(s), and H(0, t) = (0, 0), H(1, t) = (1, 0) for all t. Each H_t is a valid path from (0, 0) to (1, 0) — γ and δ are path-homotopic. More generally, any two paths between the same endpoints in ℝ² (or in any simply connected space) are homotopic. Loops are all homotopic to the constant loop; π₁(ℝ²) = {e}.

Now puncture the plane at the origin. Consider two loops based at (1, 0): the unit circle α(s) = (cos 2πs, sin 2πs) and the constant loop κ(s) = (1, 0). Are they homotopic in ℝ² ∖ {0}?

No. A homotopy H(s, t) from α to κ in ℝ² ∖ {0} would have, at each fixed t, a loop based at (1, 0) avoiding the origin. The winding number w(H_t) around the origin is integer-valued and continuous in t (the integer is a topological invariant of a loop avoiding the origin). w(α) = 1 ≠ 0 = w(κ), so there's no continuous interpolation through integers. The loops live in different homotopy classes. The fundamental group of the punctured plane is ℤ — the winding-number map is an isomorphism. Two homotopy classes: one for each integer.

Why homotopy is well-defined on loops

To verify that homotopy classes of loops form a group, the key facts are:

1. Reparametrisation. If φ : [0, 1] → [0, 1] is continuous and surjective with φ(0) = 0, φ(1) = 1, then γ ∘ φ ≃ γ via H(s, t) = γ((1 − t)φ(s) + ts).
2. Associativity up to homotopy. For three loops α, β, γ, the two parenthesisations (α · β) · γ and α · (β · γ) — both defined piecewise but with different speeds — are homotopic via a reparametrisation. So at the level of homotopy classes [α]([β][γ]) = ([α][β])[γ].
3. Identity element. The constant loop c at x₀ satisfies c · γ ≃ γ ≃ γ · c. A homotopy: slow γ down on the first half (where you stand still) and speed it up on the second half.
4. Inverses. γ̄(s) = γ(1 − s) satisfies γ · γ̄ ≃ c. A homotopy: at time t, walk along γ until parameter s = t, then walk back along γ̄ from γ(t) to γ(0).

Together these make π₁(X, x₀) into a group. Different basepoints in a path-connected space give isomorphic groups (the isomorphism is conjugation by a path joining the basepoints; it depends on the path, so the isomorphism is non-canonical).

Homotopy equivalence

Two spaces X and Y are homotopy equivalent (written X ≃ Y) if there exist continuous f : X → Y and g : Y → X with f ∘ g ≃ id_Y and g ∘ f ≃ id_X. The maps f and g are homotopy inverses; they need not be bijections.

The disc D² is homotopy equivalent to a point: f : D² → {p} is the constant map, g : {p} → D² is the inclusion at the centre, f ∘ g = id_{p}, and g ∘ f ≃ id_{D²} via the straight-line homotopy contracting D² to the centre. But D² is not homeomorphic to a point — homeomorphism is much stricter. Spaces homotopy equivalent to a point are called contractible; ℝⁿ, D^n, any convex subset of ℝⁿ are contractible.

The Möbius strip M is homotopy equivalent to its core circle S¹ (deformation retract along the width). Both have π₁ ≅ ℤ. M and S¹ are not homeomorphic — they have different dimensions — but they share every homotopy invariant.

Homotopy equivalence preserves all homotopy invariants: homotopy groups π_n, homology groups H_n, cohomology rings, characteristic classes. If you compute different values of any of these for X and Y, they are not homotopy equivalent — let alone homeomorphic.

Variants of homotopy

• Based homotopy. Maps preserving basepoints (pointed spaces); homotopies that fix the basepoint at every time t. The natural setting for π_n(X, x₀) for all n ≥ 0.
• Relative homotopy. Homotopy of maps of pairs (X, A) → (Y, B), with homotopy taking A to B at every time. Yields relative homotopy groups π_n(X, A, x₀) and the long exact sequence of a pair.
• Smooth homotopy. When X, Y are smooth manifolds, restrict H to be smooth. Whitney approximation: continuous maps between smooth manifolds are homotopic to smooth ones. Smooth homotopy gives the same homotopy categories as continuous homotopy.
• Stable homotopy. Iterated suspension X → ΣX → Σ²X… stabilises homotopy groups. π_n^s(X) is the colimit of π_{n+k}(Σ^k X). Stable homotopy theory (Adams, Quillen, May) is its own subject.
• Higher homotopy. π_n(X, x₀) classifies based maps S^n → X up to homotopy. Abelian for n ≥ 2 (Eckmann-Hilton). The computation of π_n(S^k) for n > k is famously deep and incomplete.
• Homotopy categories and ∞-categories. The collection of all spaces with maps modulo homotopy forms the homotopy category Ho(Top); replacing strict equality with higher homotopies leads to ∞-groupoids, model categories, and the homotopy type theory of Voevodsky.

Where homotopy shows up

• Algebraic topology. π_n and H_n are homotopy invariants — assigning groups to spaces that respect homotopy equivalence. The whole subject is the study of these invariants.
• Knot theory. Knots in ℝ³ are studied up to ambient isotopy (a stronger relation than homotopy), but the knot complements ℝ³ ∖ K are studied as topological spaces, and homotopy invariants — the knot group π₁(ℝ³ ∖ K) — distinguish many knots.
• Differential equations. The Conley index uses homotopy to study isolated invariant sets of flows; the index is a homotopy type, and Morse theory connects critical points to homotopy of sublevel sets.
• Robotics and motion planning. Two paths between robot configurations are homotopic iff they avoid the same obstacles in the same way; computing homotopy classes (often via PRM or RRT methods) sorts paths into qualitative families.
• Type theory. Homotopy type theory (HoTT) identifies types with spaces, terms with points, identifications with paths, and proofs with higher homotopies — Voevodsky's univalence axiom turns the homotopy interpretation into a foundation for mathematics.
• Physics: anyons and topological order. Configuration spaces of identical particles in 2D have π₁ = braid group; exchanges of anyons implement non-trivial homotopy classes — the basis of topological quantum computation proposals.

Common pitfalls

• Forgetting continuity of H. A piecewise definition that gives the right values at t = 0 and t = 1 is not a homotopy unless the family is jointly continuous in (x, t). A common error is to define H_t separately for each t without checking that t-dependence is continuous.
• Confusing homotopy with homeomorphism. A disc is homotopy equivalent to a point but not homeomorphic to one (different cardinalities). Homotopy equivalence is much coarser; it doesn't preserve dimension, compactness, or local structure.
• Confusing homotopy with isotopy in knot theory. All knots are homotopic to the unknot (in ℝ³, every embedded loop can be contracted by allowing self-crossings during the deformation). Knots are distinguished by isotopy — homotopies through embeddings — not arbitrary homotopies.
• Path-homotopy vs free homotopy. For loops, homotopy that fixes the basepoint at every time gives π₁; homotopy that lets the basepoint wander gives conjugacy classes of π₁. They differ in non-abelian fundamental groups.
• Assuming π_n distinguishes all spaces. Whitehead's theorem says a map between CW complexes is a homotopy equivalence iff it induces isomorphisms on all π_n. But two spaces having identical π_n for every n does not automatically mean they're homotopy equivalent — you need an actual map realising the isomorphisms.
• Convex combinations only work in linear spaces. The straight-line homotopy H(x, t) = (1 − t)f(x) + t g(x) requires Y to be a vector space (or a convex subset). On a sphere or torus the formula doesn't make sense; you need a different interpolation (e.g. geodesic homotopy).