Homotopies and homotopy equivalence

For continuous maps $f,g:X\to Y$, a homotopy from f to g is a continuous map $H:X\times I\to Y$, where $I=[0,1]$, with $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Each time slice is a map, but continuity in both variables together is essential. Choosing a path from f(x) to g(x) for each x need not produce a continuous family. If the deformation must fix a subspace A, impose $H(a,t)=f(a)=g(a)$ for all a and all times; this is a homotopy relative to A.

When the target is a convex subset of a real vector space, the straight-line formula joins any two continuous maps whose values lie there. Convexity ensures the entire interpolation stays in the target. The same formula can fail on a punctured plane: interpolating opposite points may pass through the forbidden origin. A useful verification checklist is therefore endpoint values, joint continuity, target membership, and any relative condition. A visually plausible deformation that leaves the prescribed space is not a homotopy of maps into that space.

A map f from X to Y is a homotopy equivalence if some g from Y to X satisfies $gf\simeq\mathrm{id}_X$ and $fg\simeq\mathrm{id}_Y$. The composites need not equal the identities. A homeomorphism supplies such an equivalence with exact inverse composites, but an equivalence can forget contractible thickness or branches. Contractibility means the identity is homotopic to a constant map, equivalently that the nonempty space is homotopy equivalent to a point. Path connectedness alone is far weaker: it supplies paths between points without choosing the coherent contraction required by a homotopy.

A retraction $r:X\to A$ fixes A pointwise, so the inclusion i obeys $ri=\mathrm{id}_A$. A deformation retraction additionally deforms the identity of X to ir while keeping A fixed throughout. It therefore makes i and r homotopy inverses. An arbitrary retraction need not do this: the remaining composite ir may fail to be homotopic to the identity. The fixed-subspace condition also distinguishes a strong deformation retraction from a deformation that merely ends on A. Naming the maps and the two composite conditions prevents these notions from being conflated.

Composing homotopies with continuous maps preserves their endpoint identities, which makes homotopy equivalence an equivalence relation. Topological invariants defined through homotopy classes, such as homotopy groups with appropriate basepoint bookkeeping, consequently agree for homotopy-equivalent spaces. This does not preserve every geometric quantity: lengths, areas, curvature, and even manifold dimension can change. A point and a closed interval are homotopy equivalent, while no homeomorphism identifies their underlying spaces. The desired invariant determines whether this coarser comparison is appropriate.

A deformation retraction can simplify a calculation before algebra enters. A punctured plane retracts radially onto a circle, a cylinder retracts along its interval coordinate, and a finite tree can contract by removing its terminal branches. Each example still requires a well-defined global formula or a compatible cellwise construction. In particular, a space with a hole cannot be declared contractible merely because a small piece of it is. Later the fundamental group will obstruct some proposed contractions, while higher homotopy groups detect holes invisible to loops.