Homotopy Theory — Mathematics Knowledge System

Overview

Homotopy theory studies continuous deformations of maps and spaces. Two maps are homotopic if one can be continuously deformed into the other. This notion captures the idea that certain topological properties are preserved under continuous deformation, giving rise to a rich classification theory for topological spaces up to homotopy equivalence.

Intuition

Think of two rubber band shapes on a flat table. If you can slide one shape into the other without lifting it or tearing it, they are homotopic. A circle and a square are homotopic (same rubber band shape), but a circle and a figure-eight are not (one has one hole, the other has two). A space is contractible if it can be continuously shrunk to a point — like a disk but unlike a circle. Homotopy equivalence is the topological version of 'same shape' that respects continuous deformation but not rigid geometry.

Formal Definition

Definition

A homotopy between two continuous maps f, g: X → Y is a continuous map H: X × [0,1] → Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x. We write f ≃ g. A map f: X → Y is a homotopy equivalence if there exists g: Y → X such that g∘f ≃ id_X and f∘g ≃ id_Y. A space X is contractible if the identity map id_X is homotopic to a constant map. A subspace A ⊆ X is a deformation retract if there is a homotopy H: X × [0,1] → X with H(x,0) = x, H(x,1) ∈ A for all x, and H(a,t) = a for all a ∈ A.

H: X \times [0,1] \to Y,\quad H(x,0)=f(x),\quad H(x,1)=g(x)

Homotopy

f \simeq g \iff \exists\, H \text{ homotopy from } f \text{ to } g

Homotopic maps

X \simeq Y \iff \exists\, f:X\to Y,\, g:Y\to X:\; g\circ f \simeq \mathrm{id}_X,\; f\circ g \simeq \mathrm{id}_Y

Homotopy equivalence

X \text{ contractible} \iff \mathrm{id}_X \simeq c_x \text{ for some constant map } c_x

Contractible

Notation

Notation	Meaning
$f \simeq g$	f is homotopic to g
$X \simeq Y$	X and Y are homotopy equivalent
$[X,Y]$	Set of homotopy classes of maps from X to Y
$[f]$	Homotopy class of the map f
$H_t$	The map x ↦ H(x,t) for fixed t

Theorems

Theorem 1: Whitehead's Theorem

If f: X → Y is a map between CW complexes that induces isomorphisms on all homotopy groups π_n for n ≥ 0, then f is a homotopy equivalence.

Theorem 2: Homotopy Invariance of Homology

If f: X → Y is a homotopy equivalence, then the induced maps f_*: H_n(X) → H_n(Y) are isomorphisms for all n ≥ 0.

Theorem 3: Contractible Spaces are Acyclic

If X is contractible, then H_n(X) = 0 for all n > 0 and H_0(X) ≅ Z.

Worked Examples

1
We need to find a homotopy between the identity map on R^n and a constant map.
2
Define H: R^n × [0,1] → R^n by H(x, t) = (1-t)x.
$H(x,t) = (1-t)x$
3
At t=0: H(x,0) = x = id_{R^n}(x). At t=1: H(x,1) = 0 = c_0(x).
4
H is continuous since it is a polynomial in t and the coordinates of x. Therefore id_{R^n} ≃ c_0, so R^n is contractible.

✓ Answer

The straight-line homotopy H(x,t) = (1-t)x contracts R^n to the origin.

Practice Problems

Mediumproof writing

Prove that homotopy of maps is an equivalence relation.

Mediumproof writing

Prove that any convex subset of R^n is contractible.

Common Mistakes

Common Mistake

Homotopy equivalence is the same as homeomorphism.

Homeomorphism requires a continuous bijection with continuous inverse, which is strictly stronger. Homotopy equivalence only requires continuous maps in both directions whose compositions are homotopic to identities. For example, R^1 ≃ {pt} but they are not homeomorphic.

Common Mistake

A retract is always a deformation retract.

A retract r: X → A satisfies r|_A = id_A but the inclusion need not be homotopic to id_X. A deformation retract additionally requires the inclusion i: A → X to satisfy i∘r ≃ id_X via a homotopy fixing A.

Quiz

Historical Background

Homotopy theory emerged from Poincaré's foundational work in the 1890s, where he introduced the fundamental group as a way to distinguish topological spaces. The systematic study of homotopy equivalences developed through the 20th century, with major contributions from Hurewicz, who introduced higher homotopy groups in 1935, and J.H.C. Whitehead, who developed the theory of CW complexes and proved his celebrated theorem. The modern axiomatic treatment through model categories was developed by Quillen in 1967.

1895
Poincaré introduces the fundamental group in Analysis Situs
Poincaré
1935
Hurewicz defines higher homotopy groups π_n
Hurewicz
1949
Whitehead proves the Whitehead theorem on homotopy equivalences
J.H.C. Whitehead
1967
Quillen introduces model categories for axiomatic homotopy theory
Quillen

Summary

A homotopy between maps f, g: X → Y is a continuous interpolation H: X × [0,1] → Y.
Two spaces are homotopy equivalent if they are related by maps whose compositions are homotopic to identities.
A contractible space is homotopy equivalent to a point; R^n and convex subsets are examples.
Homotopy invariants such as homology groups, cohomology groups, and homotopy groups are unchanged by homotopy equivalences.
Whitehead's theorem: for CW complexes, a map inducing isomorphisms on all homotopy groups is a homotopy equivalence.

References

BookHatcher, A. Algebraic Topology. Cambridge University Press, 2002. Chapter 0.
BookMay, J.P. A Concise Course in Algebraic Topology. University of Chicago Press, 1999.
WebsiteWikipedia: Homotopy