Deformation retracts and homotopy equivalence

Try

The punctured plane contains many radial distances but only one angular loop direction. Move each point radially to the unit circle without ever crossing the missing origin. The formula H(x,s)=((1−s)+s/||x||)x performs this continuously and fixes points already on the circle. This reduces a two-dimensional space to a one-dimensional loop model.

Predict

Predict whether the same formula works at the origin. It is undefined there, exactly where a radial direction has no unique value. Predict whether every retraction comes from such a deformation. A disconnected two-point space retracts onto either point, but cannot move the other point continuously to it. Retraction and deformation retraction are different requirements.

Experiment

Track only the radius under the punctured-plane deformation.

Initial radius	Radius at s=1/2	Final radius
1/2	3/4	1
1	1	1
3	2	1

Every intermediate radius is positive. The middle row stays fixed throughout. Draw two rays and verify that their angles never change. These observations check the image and relative conditions; the explicit formula supplies joint continuity.

Understand

A retraction r:X→A of a subspace A satisfies r(a)=a for every a in A. A strong deformation retraction additionally supplies H:X×[0,1]→X from id_X to i∘r with H(a,s)=a throughout, where i is inclusion. The word strong records that fixed-subspace condition; some authors use deformation retraction with this condition built into the definition. We state it explicitly to avoid a convention mismatch.

The radial formula above is defined on every nonzero vector. Its coefficient is positive for all s, so it stays in the punctured plane. It begins at x, ends at its radial projection and fixes unit vectors. The inclusion and projection are therefore homotopy inverses. A similar formula retracts an annulus onto any intermediate concentric circle. The cylinder S¹×R retracts onto S¹×a single height by linearly changing only the second coordinate.

More generally, maps f:X→Y and g:Y→X form a homotopy equivalence if g∘f is homotopic to id_X and f∘g to id_Y. Based homotopy inverses induce inverse group homomorphisms immediately. If the homotopies move basepoints, their tracks supply basepoint-change isomorphisms. The induced composites are then identities up to conjugation, which remains an automorphism. It follows that f_* is injective and surjective after the appropriate basepoint identifications, hence an isomorphism. The conclusion is not a literal equality of arbitrary point-based formulas.

A homotopy equivalence need not be a homeomorphism: the punctured plane and circle have different local dimensions but equivalent loop structure. Conversely a retraction alone only yields an injection from the subspace group and a surjection back, not necessarily an isomorphism. The discrete two-point example shows why no contraction follows merely from the retraction identity.

Apply

For a star-shaped subset of Euclidean space with center a, straight-line contraction to a stays inside the set and fixes a. This proves contractibility, even if the set is not convex. Explain which segment condition is actually required. When a proposed contraction passes through a missing point or leaves the space, identify that failure before using it to compute any group.

Continue

A well-chosen deformation retract makes later computations simpler while preserving the justified invariant. Keep topology and homotopy type separate. Next, covering maps will provide a different simplification: loops in a complicated base can become ordinary paths in a space with separated local sheets.

Continue to Covering maps and local sheets.