A coffee cup and a doughnut are often called the same shape: one can be kneaded into the other like clay without closing the hole of the handle. Homotopy turns that picture into a statement about maps [1].
Take topological spaces , and continuous maps . A homotopy from to is a continuous map with and for every [1,2].
Read as time and write . Then , , and the maps in between form a family that carries into . When such an exists, and are homotopic, written .
It is not enough for each to be continuous by itself. The definition asks for to be continuous on the whole of , that is, continuous in as well as in .
Without that condition any two maps would be joined: keep up to and switch to from then on. Every is continuous, but the family jumps at one instant, and a jump is exactly what a deformation is not allowed to do.
A homotopy is therefore one continuous map on the product space, not a list of continuous maps. In the figure the left family passes through every intermediate position, while the right one only swaps for .
Homotopy is an equivalence relation. Every map is homotopic to itself through , and running a homotopy backwards as gives symmetry [2].
For transitivity, take from to and from to and run them one after the other: at double speed for , then on the second half. The pieces agree at , so the pasting lemma makes the whole continuous.
The equivalence classes are the homotopy classes. If and , then , so composition is well defined on classes [1].
With spaces as objects and homotopy classes as morphisms this gives a category, the homotopy category. Two spaces are isomorphic there exactly when they are homotopy equivalent, which is the next notion below.
Let be convex and let be any continuous maps. Put . Each value lies on the segment from to , and convexity keeps that segment inside [1].
This is the straight-line homotopy. It shows that any two maps into a convex set are homotopic, so maps into form a single homotopy class.
In the figure the two thin paths are and and the bold one is . Every point slides along its own segment, and the endpoints, where and agree, do not move at all.
A map is null-homotopic when it is homotopic to a constant map [1]. For a loop this has a concrete meaning: the loop is null-homotopic exactly when it extends to a continuous map on the disk it bounds.
In the plane every loop can be filled in, so every loop shrinks to a point. Remove one point and a loop around the gap can no longer be filled; however it is deformed, it keeps catching on the missing point.
The identity is not null-homotopic, yet the inclusion of the same circle into is. Whether a loop is null-homotopic depends on the space it maps into, not on the circle alone [4].
Spaces and are homotopy equivalent when there are maps and with and [1,2]. They are then said to have the same homotopy type.
For a homeomorphism the two composites equal the identities. Homotopy equivalence only asks them to be homotopic to the identities, so every homeomorphism is a homotopy equivalence and not the other way round.
The disk and a point are homotopy equivalent without being homeomorphic: one has infinitely many points and the other has one. The disk shrinks onto its centre through .
Being coarser makes the relation easier to decide, and many topological invariants turn out to depend only on the homotopy type.
For , a retraction is a continuous map that is the identity on [3]. If moreover , where is the inclusion, then is a deformation retract of : the whole of can be pushed continuously onto .
If the points of must stay put during the entire deformation, is a strong deformation retract.
Take the annulus in the plane, with its inner boundary. The homotopy pushes every point radially inward and lands on at , while the inner circle never moves.
The same formula works on , the plane with the origin removed. Every point slides along its ray onto the unit circle, so is a strong deformation retract of the punctured plane [3].
Circles outside move inward and circles inside move outward, and only the unit circle stays where it is. Had the origin not been removed, the whole plane would collapse to one point; the single missing point is what stops that.
In general is a strong deformation retract of . Removing one point leaves exactly the information of a sphere, so questions about punctured spaces come down to questions about spheres.
The annulus , the punctured plane and the Möbius band all deformation retract onto a circle; the band does so by shrinking its width to zero around its central circle. All three have the homotopy type of .
They look quite different, yet every homotopy invariant agrees on them. For example, the fundamental group of each one is , the same as .
The annulus and the Möbius band are not homeomorphic: the annulus has two boundary circles and the band has only one. The difference is real, but it is exactly the kind of difference that homotopy equivalence cannot see.
For any space the cone is : the cylinder over with the end squeezed to a single point, the apex.
The homotopy slides every point along its segment to the apex, so is contractible whatever is [4].
Since sits inside as the other end, every space embeds in a contractible space. The larger space therefore tells nothing about ; the information lies in how itself is put together.
A space is contractible when it is homotopy equivalent to a point, or equivalently when its identity map is null-homotopic [4]. The space , convex sets and star-shaped sets are all contractible, each shrinking onto a centre along straight segments.
A contractible space has trivial homotopy groups and trivial homology, so most algebraic invariants see nothing in it. The converse needs care: the Warsaw circle has all its homotopy groups trivial and is still not contractible, because it behaves badly at a small scale.
For CW complexes the two notions agree. This follows from Whitehead's theorem: a map between CW complexes that induces isomorphisms on all homotopy groups is a homotopy equivalence.
A homotopy relative to is one that leaves the points of where they are: for every and every [1].
For paths, consists of the two endpoints, and a homotopy relative to them is called a path homotopy. The fundamental group is built from exactly this kind of homotopy.
The endpoints have to be held. If they were free, every loop could be pulled back along itself to its starting point and all loops would fall into one class. With the endpoints fixed, a path passing above a hole cannot be moved to a path passing below it.
Homotopy equivalent spaces share their homotopy groups , their homology and cohomology, their number of path components and, for finite CW complexes, their Euler characteristic .
Dimension, compactness and the number of boundary components are not kept. The disk and the point already show this: drops from 2 to 0 while stays the same.
What homotopy throws away is, roughly, thickness. Holes and the way pieces are joined survive, while size and dimension do not, and giving that much up is what lets algebra in. The fundamental theorem of algebra, Brouwer's fixed point theorem and the Borsuk-Ulam theorem are all proved by showing that some homotopy class cannot vanish [5,6].