Homotopy
When should two continuous maps count as the same because one can be deformed into the other?
What changes when transformations themselves become objects of comparison?
Topology began by relaxing geometry. Distance and angle stopped being primary, while continuity and open-set structure remained. Homeomorphism then gave a standard of sameness for spaces: two spaces are topologically the same when each can be continuously and reversibly translated into the other.
Compactness and connectedness used that standard to produce invariants of spaces. They asked what survives when the space is stretched, bent, or continuously relabeled.
Homotopy shifts attention one level higher.
Instead of asking only when two spaces are the same, it asks when two continuous maps are the same up to continuous deformation.
f0 : X -> Y
f1 : X -> Y
can f0 be continuously deformed into f1?
This is a new structural move. The objects being compared are no longer just points or spaces. They are transformations between spaces.
Imagine a loop drawn on a surface. The loop is a continuous map from a circle into the space:
S1 -> X
If the loop can slide across the surface without tearing, jumping, or leaving the space, topology may treat the original loop and the final loop as the same kind of loop. The exact route changes. The continuous deformation is what matters.
loop at time 0
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| deform continuously
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loop at time 1
A homotopy makes this precise.
Given two continuous maps f0, f1: X -> Y, a homotopy from f0 to f1 is a continuous map
H: X x [0, 1] -> Y
such that
H(x, 0) = f0(x)
H(x, 1) = f1(x)
for every point x in X.
The extra interval [0, 1] is a deformation parameter. At time 0, the homotopy is the first map. At time 1, it is the second map. At every intermediate time t, the rule
x |-> H(x, t)
is another continuous map from X to Y.
X ----f0----> Y time 0
X ----ft----> Y time t
X ----f1----> Y time 1
So a homotopy is a continuous family of maps. If the family could jump, any map might be replaced by a completely unrelated map in one sudden step. Homotopy keeps the deformation inside topology: change is allowed, but it must happen continuously.
This creates a new equivalence relation on maps. A map is homotopic to itself by the constant deformation. If f0 deforms into f1, then running the deformation backward deforms f1 into f0. If f0 deforms into f1, and f1 deforms into f2, the two deformations can be run one after the other to deform f0 into f2.
f0 ~ f1
f1 ~ f2
-------
f0 ~ f2
The resulting equivalence classes are homotopy classes.
continuous maps X -> Y
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| identify maps connected by deformation
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homotopy classes
This is topology taking its own idea seriously. Earlier, equality of spaces gave way to homeomorphism. Now equality of maps gives way to homotopy.
spaces:
same up to homeomorphism
maps:
same up to homotopy
The difference is important. Homeomorphism is a reversible exact match of topological structure. Homotopy is a weaker relation. It allows maps to move through a continuous family, even when the intermediate maps are not reversible and even when the original maps are not homeomorphisms.
This weaker relation detects different information.
Consider loops in the plane. A loop can usually be contracted continuously to a point if the whole disk inside it is available. On a punctured plane, a loop that winds around the missing point cannot be contracted without crossing the missing point. The missing point is not part of the space, so the deformation is blocked.
plane:
loop contracts to point
punctured plane:
loop around hole cannot contract
The obstruction is not length, curvature, or angle. It is topological. The loop remembers how it sits around a hole.
This is why homotopy leads naturally toward the study of holes. Connectedness can say whether a space is one piece. Homotopy can ask whether loops inside that piece can be shrunk, moved past obstacles, or transformed into one another.
connectedness:
is the space one piece?
homotopy:
what deformations are possible inside that piece?
The same idea applies beyond loops. Paths can be homotopic relative to their endpoints. Maps from spheres can record higher-dimensional holes. Families of maps can reveal structure that open sets alone made difficult to see directly.
At this point, topology is no longer only classifying spaces by homeomorphism. It is studying the behavior of maps into and out of spaces, and the deformation classes of those maps.
Homotopy also gives a more flexible notion of sameness for spaces.
Two spaces are homotopy equivalent when there are continuous maps
X ----f----> Y
Y ----g----> X
such that g . f is homotopic to the identity map on X, and f . g is homotopic to the identity map on Y.
g . f ~ id_X
f . g ~ id_Y
This resembles homeomorphism, but it is weaker. A homeomorphism has actual inverse maps. A homotopy equivalence has inverse maps only up to deformation.
homeomorphism:
inverse exactly
homotopy equivalence:
inverse up to continuous deformation
A solid disk and a point are not homeomorphic: one has many points and the other has one. But the disk is homotopy equivalent to a point because the whole disk can contract continuously to any chosen point inside it. Homotopy ignores the extra spread of the disk when that spread carries no essential deformation obstruction.
A circle is different. It is not homotopy equivalent to a point. A loop around the circle cannot be contracted inside the circle itself. There is no interior disk available inside the space to fill it in.
disk -> point:
homotopy equivalent
circle -> point:
not homotopy equivalent
Homotopy equivalence therefore treats some geometric size and shape information as disposable while retaining information about holes and deformation behavior.
This creates another layer of invariants. A property is a homotopy invariant when it is preserved under homotopy equivalence. Connectedness is preserved under homotopy equivalence in the usual spaces considered here. More refined invariants, such as the fundamental group, are built specifically to record how loops deform.
The central pattern is familiar by now:
choose allowed transformations
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v
define equivalence
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study what survives
For rigid geometry, the allowed transformations were isometries. For topology, homeomorphisms preserved open-set structure exactly. For homotopy theory, continuous deformation becomes the relevant comparison. This completes the topology branch at the level of structural orientation. We have seen spaces, open sets, continuous maps, homeomorphisms, compactness, connectedness, and homotopy. Each step weakened unnecessary structure while preserving a precise kind of mathematical control.
distance and angle
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v
open-set structure
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v
homeomorphism of spaces
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v
homotopy of maps
The objects are continuous maps between topological spaces. The morphisms between those maps are continuous deformations, expressed as homotopies H: X x [0, 1] -> Y. What composes are deformations and continuous maps, with homotopy behaving as an equivalence relation on maps. The invariant is deformation class: what remains unchanged when maps move continuously through other maps. The defining relation is f0 ~ f1 when there exists a continuous homotopy from f0 to f1. Equality is replaced by sameness up to deformation. What remains outside the frame is the repeated pattern behind all these examples: objects, structure-preserving maps, composition, equivalence, and invariants.
Homotopy studies when transformations are the same up to continuous deformation.
This completes the first sweep through mathematical structure. The sequence began with numbers and operations, moved through sets, relations, order, algebra, analysis, and topology, and ended by treating deformations themselves as objects of study.
The next step is to stop moving from one kind of structure to another and instead ask why the same pattern kept appearing. Numbers had operations. Sets had functions. Algebra had homomorphisms. Vector spaces had linear maps. Topological spaces had continuous maps. Homotopy then treated deformations between maps as structure too.
What is the common structure behind objects, maps, and composition themselves?