List of Homotopy Theories

Introduction

This article lists the homotopy-theoretic objects and theories of Parts I to III, grouped by the layer at which they act: the fundamental group and the covering spaces, which are the theory of loops and of the maps of an interval; the higher homotopy groups, the fibrations and the cofibrations, which are the theory of the maps of spheres and of the lifting and extension properties; the CW complexes, on which the homotopy invariants are computable; and the model categories, the higher categories and the stable theory, which are the abstract framings in which the same constructions are performed in other categories. The theorems that organise the subject — van Kampen, Hurewicz, Whitehead, Blakers–Massey, Freudenthal, cellular approximation — are tabulated with the layer that proves them.

Every entry points to the article that introduces the object or the theorem. The article introduces nothing and proves nothing: it records the statement and the hypothesis of the introducing article, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the spaces on which the theorems apply it lists the spaces on which they fail — the fundamental group, which is non-abelian and has no higher-degree analogue with coefficients in a non-abelian group; the Warsaw circle, which has the Čech cohomology of a circle without being homotopy equivalent to one; the spaces that admit no universal cover for want of semilocal simple connectivity; the weak homotopy equivalences that are not homotopy equivalences in the absence of a CW structure — each with the failure named and the article that records it.

The Fundamental Group and Covering Spaces

The fundamental group is the first algebraic invariant of a space, and the covering spaces are the spaces whose fundamental group theory mirrors the subgroup structure of the group.

Object or theorem	The statement or the structure	Introduced in
the fundamental group $\pi_1(X,x_0)$	path-homotopy classes of loops at $x_0$, with concatenation; functorial, with basepoint change by conjugation	The Fundamental Group and Covering Spaces
homotopy and homotopy equivalence	$f \simeq g$ through a homotopy $F : X \times I \to Y$; contractible spaces	The Fundamental Group and Covering Spaces
the covering space $p : E \to X$	a surjection locally a product; the path and homotopy lifting properties hold	The Fundamental Group and Covering Spaces
the classification of coverings	connected coverings of a nice space correspond to the subgroups of $\pi_1(X,x_0)$, up to isomorphism	The Fundamental Group and Covering Spaces
the universal cover $\tilde X$	the simply connected covering; $\pi_1(X,x_0) \cong \operatorname{Deck}(\tilde X/X)$	The Fundamental Group and Covering Spaces
deck transformations $\operatorname{Deck}(E/X)$	the homeomorphisms over $X$; the Galois correspondence of a covering	The Fundamental Group and Covering Spaces
van Kampen's theorem	$\pi_1(A \cup B)$ is the amalgamated product of $\pi_1(A)$ and $\pi_1(B)$ over $\pi_1(A \cap B)$ under the stated hypotheses	The Fundamental Group and Covering Spaces
the computation of $\pi_1(S^1)$	$\pi_1(S^1,1) \cong \mathbb{Z}$, generated by $\omega(s) = e^{2\pi i s}$	The Fundamental Group and Covering Spaces
the free group $F_n$	$\pi_1$ of a wedge of $n$ circles; the free product $A \ast B$ of the van Kampen theorem	The Fundamental Group and Covering Spaces; Combinatorial Group Theory

The Higher Homotopy Groups, Fibrations and Cofibrations

The higher homotopy groups are the maps of spheres, and the fibrations and cofibrations are the maps with the lifting and extension properties that make the groups computable.

Object or theorem	The statement or the structure	Introduced in
the homotopy groups $\pi_n(X,x_0)$	based homotopy classes $S^n \to X$; abelian for $n \geq 2$; $\pi_0$ is the set of path components	Homotopy Groups and Fibrations
the loop space $\Omega X$ and the suspension $\Sigma X$	$\pi_n(X) \cong \pi_{n-1}(\Omega X)$; the path–loop fibration $PX \to X$ with fibre $\Omega X$	Homotopy Groups and Fibrations
a fibration $F \to E \to B$	the homotopy lifting property; the long exact sequence $\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots$	Homotopy Groups and Fibrations
the Hurewicz theorem	the first nonvanishing homotopy and homology groups agree, $h_n : \pi_n(X) \to H_n(X;\mathbb{Z})$, for a simply connected space	Homotopy Groups and Fibrations
the Whitehead theorem	a map of simply connected CW complexes inducing isomorphisms on all homotopy groups is a homotopy equivalence	Homotopy Groups and Fibrations
the Blakers–Massey theorem	homotopy excision: a union $A \cup B$ is compared with $A$ and $B$ through their intersection, and the connectivity of the comparison map follows; it gives the map $S^n \to \Omega S^{n+1}$ connectivity $n-1$	Homotopy Groups and Fibrations
the Freudenthal suspension theorem	$\pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ is an isomorphism for $n > k+1$	Homotopy Groups and Fibrations; Stable Homotopy Theory
the Hopf fibrations	$S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$, $S^7 \to S^{15} \to S^8$ over $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$; the Hopf map $\eta$ generates $\pi_3(S^2)$	Homotopy Groups and Fibrations
a weak homotopy equivalence and an $n$-equivalence	isomorphisms on all $\pi_n$, all basepoints; the CW approximation that repairs the failure without a CW structure	Homotopy Groups and Fibrations; CW Complexes and Cellular Approximation
an Eilenberg–MacLane space $K(\pi,n)$	the space with the single nonvanishing homotopy group $\pi$ in degree $n$; represents $H^n(-;\pi)$	Classifying Spaces and Cohomology Operations
the Postnikov tower and the $k$-invariants $k_n$	the successive fibrations whose fibres are Eilenberg–MacLane spaces; the obstruction theory it carries	Classifying Spaces and Cohomology Operations

The CW Complexes and the Cofibrations

The CW complexes are the spaces assembled from discs, and the homotopy extension property is the cofibration that makes the assembly well behaved.

Object or theorem	The statement or the structure	Introduced in
a CW complex $X$	a space with a cell filtration $X^n$ and the weak topology on the cells; Hausdorff and paracompact	CW Complexes and Cellular Approximation
the attaching maps and the cells	$\varphi_\alpha : D^n \to X$, with open cell $e^n_\alpha = \varphi_\alpha(\mathring D^n)$	CW Complexes and Cellular Approximation
the homotopy extension property (HEP)	the extension of a homotopy from $A$ to $X$; for a relative CW pair, equivalently the inclusion is a cofibration	CW Complexes and Cellular Approximation
the cellular approximation theorem	every map of CW complexes is homotopic to a cellular one; $\pi_n(S^n) \cong \mathbb{Z}$ and $\pi_k(S^n) = 0$ for $k < n$	CW Complexes and Cellular Approximation
the cellular chain complex	free on the $n$-cells, with $\partial_n e^n_\alpha = \sum_\beta \deg(q_\beta\varphi_\alpha)e^{n-1}_\beta$; its homology is the singular homology	CW Complexes and Cellular Approximation
the degree of a map of spheres	$\deg f$ with $f_* = \deg f\cdot\mathrm{id}$ on $H_n(S^n;\mathbb{Z})$	Degree Theory and the Brouwer Fixed Point Theorem
the Euler characteristic	$\chi(X) = \sum_n(-1)^n\#\mathcal{E}_n$, equal to the alternating sum of the Betti numbers	CW Complexes and Cellular Approximation
the quotient and smash product $X/A$, $X \wedge Y$, $\Sigma X$	the collapse of a subcomplex, the smash product of based complexes and the reduced suspension	CW Complexes and Cellular Approximation
the simplicial complex and the nerve $\lvert N(\mathcal{U})\rvert$	a combinatorial model, with the nerve of a good cover homotopy equivalent to the space	Simplicial and Singular Homology; Čech Cohomology

The Model Categories and the Abstract Homotopy Theory

The constructions of the preceding sections depend only on three classes of maps, and this is the abstraction under which they are performed in other categories.

Object or theory	The structure it supplies	Introduced in
a model category $(\mathcal{C},\mathcal{W},\mathcal{C}\mathrm{of},\mathcal{F}\mathrm{ib})$	a bicomplete category with weak equivalences, cofibrations and fibrations satisfying the lifting and factorisation axioms	Model Categories and Homotopy Theory
the homotopy category $\operatorname{Ho}(\mathcal{C})$	the localisation $\mathcal{C}[\mathcal{W}^{-1}]$, computed as $\pi\operatorname{Hom}(QX,RY)$	Model Categories and Homotopy Theory
cylinder and path objects	the intrinsic replacements of $X \times I$ and of the path space	Model Categories and Homotopy Theory
Quillen adjunctions and derived functors $\mathbb{L}F$, $\mathbb{R}G$	the left and right derived functors of a Quillen pair; homotopy limits and colimits	Model Categories and Homotopy Theory
the model structure on simplicial sets	the Kan complexes and the Kan fibrations; the singular functor and the geometric realisation	Model Categories and Homotopy Theory
the Dold–Kan correspondence	the equivalence of simplicial $R$-modules with nonnegatively graded chain complexes	Model Categories and Homotopy Theory
Bousfield localisation $L_S\mathcal{C}$	the localisation at a set of morphisms; the model-categorical form of the stable localisations	Model Categories and Homotopy Theory
an $\infty$-category	a quasi-category or complete Segal space; the mapping spaces and the homotopy category	Higher Algebra and Higher Categories
operads and their algebras	symmetric sequences with substitution, the operads $\mathcal{A}ss$, $\mathcal{C}om$, $\mathcal{L}ie$; $A_\infty$ and $E_n$ algebras	Operads; Higher Algebra and Higher Categories
a stable $\infty$-category	a pointed $\infty$-category in which $\Sigma$ and $\Omega$ are inverse equivalences and the homotopy category is triangulated	Higher Algebra and Higher Categories
the stable homotopy category $\mathcal{SH}$	the homotopy category of spectra, triangulated and symmetric monoidal under the smash product	Stable Homotopy Theory

The Stable Theory and the Generalised Invariants

The stable theory is the theory of the objects whose homotopy groups are the stable stems, and it is the setting in which the generalised (co)homology theories become representable.

Object or theory	The structure it supplies	Introduced in
a spectrum $E$ and its homotopy groups	$\pi_k(E) = \operatorname{colim}_n\pi_{k+n}(E_n)$; $\Sigma^\infty X$ and the sphere spectrum $\mathbb{S}$	Stable Homotopy Theory
the stable stems $\pi_k^s$	the colimits of $\pi_{n+k}(S^n)$ under suspension; the $J$-homomorphism maps the homotopy of $SO(n)$ into them	Stable Homotopy Theory
the representability of (co)homology	every generalised theory is represented by a spectrum, with the Eilenberg–MacLane spectra $H\pi$ for the ordinary theories	Stable Homotopy Theory
the Adams spectral sequence	$E_2^{s,t} = \operatorname{Ext}^{s,t}_{\mathcal{A}_p}(\mathbb{F}_p,H^*(E;\mathbb{F}_p))$ converging to $\pi_{t-s}(E)\otimes\mathbb{Z}_p$	Stable Homotopy Theory; Classifying Spaces and Cohomology Operations
the Atiyah–Hirzebruch spectral sequence	$E^2_{p,q} = H_p(X;E_q)$ converging to $E_{p+q}(X)$	Stable Homotopy Theory
the classifying space $BG$	the base of the universal bundle $EG \to BG$, with $[X,BG] = \operatorname{Prin}_G(X)$	Classifying Spaces and Cohomology Operations
the Steenrod algebra $\mathcal{A}_p$	the stable cohomology operations, generated by $\mathrm{Sq}^i$ and $P^i$ subject to the Adem relations	Classifying Spaces and Cohomology Operations
topological $K$-theory and cobordism	the generalised cohomology theories represented by $KU$, $KO$ and $MU$	Topological K-Theory; Cobordism and Surgery Theory
the chromatic filtration	the localisations at the Morava $K$-theories $K(n)$ and the type stratification of finite spectra	Stable Homotopy Theory

The Theorems

The theorems that the scope line names are collected here with the layer that proves them and with the hypothesis each needs.

Theorem	Its hypothesis and its content	Introduced in
van Kampen's theorem	the space is the union of open path-connected sets with path-connected intersection; the fundamental group is the amalgamated product	The Fundamental Group and Covering Spaces
Hurewicz's theorem	the space is $(n-1)$-connected; the first nonvanishing homotopy and homology groups agree through $h_n$	Homotopy Groups and Fibrations
Whitehead's theorem	the complexes are CW and simply connected; an isomorphism on all homotopy groups is a homotopy equivalence	Homotopy Groups and Fibrations
Blakers–Massey	$X$ is the union of subcomplexes $A$ and $B$ with connected intersection; the map $A \cup B \to X$ is highly connected	Homotopy Groups and Fibrations
Freudenthal's theorem	$n > k+1$; the suspension homomorphism is an isomorphism, so the stable stem is defined	Homotopy Groups and Fibrations; Stable Homotopy Theory
the cellular approximation theorem	the spaces are CW complexes; every map is homotopic to a cellular one	CW Complexes and Cellular Approximation
the classification of coverings	$X$ is path-connected, locally path-connected and semilocally simply connected; the coverings correspond to the subgroups	The Fundamental Group and Covering Spaces

Non-examples and Warnings

Object	Why the expected statement fails	Introduced in
the fundamental group $\pi_1(X,x_0)$	it is not abelian, and there is no higher analogue with coefficients in a non-abelian group; the abelianisation is $H_1$	The Fundamental Group and Covering Spaces; Homotopy Groups and Fibrations
the higher homotopy groups $\pi_n(S^k)$	they are not computable from a chain complex and are not known in closed form; there is no excision for them	Homotopy Groups and Fibrations
the Warsaw circle	it has the Čech cohomology of a circle and is not homotopy equivalent to one; the shape-theoretic invariant detects it	Continuum Theory
the spaces without a universal cover	the classification of coverings fails when the space is not semilocally simply connected; the Hawaiian earring is the standard example	The Fundamental Group and Covering Spaces
a weak homotopy equivalence	without a CW structure it need not be a homotopy equivalence; the Whitehead theorem needs the CW hypothesis	Homotopy Groups and Fibrations
a category with weak equivalences	it need not be a model category, and the homotopy category is then not computable from the three classes alone	Model Categories and Homotopy Theory
the Hopf fibration over $\mathbb{O}$	the octonionic fibration $S^7 \to S^{15} \to S^8$ exists; there is no such fibration in the next Cayley–Dickson step, by the theorem of Adams	Homotopy Groups and Fibrations; Octonion Algebra

Objects that a reader may expect to find in a list of homotopy theories, and does not.

Object	Why it is not listed	Introduced in
shape theory and Čech homotopy	the corpus records the shape-theoretic examples, the Warsaw circle among them, without developing the shape category	Continuum Theory
étale homotopy theory	the homotopy theory of a scheme, recorded with the étale site and not as a topological theory	Schemes; Sheaves on Sites
motivic homotopy theory	not introduced by the corpus	—
obstruction theory as a separate article	the obstruction classes and the Postnikov invariants are recorded with the classifying spaces and the cohomology operations	Classifying Spaces and Cohomology Operations
the homotopy groups of a Lie group	recorded with the Lie groups and the homogeneous spaces, not in the general homotopy list	Homology of Classical Groups and Homogeneous Spaces

Summary

This article has listed the homotopy theories of the corpus: the fundamental group and the covering spaces, with van Kampen's theorem and the classification of the coverings; the higher homotopy groups, the fibrations and the cofibrations, with the Hurewicz, Whitehead, Blakers–Massey and Freudenthal theorems; the CW complexes, with the homotopy extension property, the cellular approximation theorem and the cellular chain complex; the model categories, the $\infty$-categories and the operads, which carry the constructions into other categories; and the stable theory, with the spectra, the stable stems, the classifying spaces and the Steenrod operations. Beside the examples stand the non-examples: the non-abelian fundamental group, the homotopy groups that resist computation, the Warsaw circle that the shape invariant detects and homotopy does not, the spaces without a universal cover, and the weak equivalences that need the CW hypothesis. The list introduces and proves nothing; it is the index of the homotopy theory of the corpus.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$I = [0,1]$	the closed unit interval
$\pi_1(X,x_0)$, $\pi_n(X,x_0)$, $\pi_0(X)$	the fundamental group, the higher homotopy groups, the set of path components
$f \simeq g$, $X \simeq Y$	homotopic maps; homotopy equivalent spaces
$p : E \to X$, $\tilde X$	a covering space; the universal cover
$\operatorname{Deck}(E/X)$	the deck transformations
$\Omega X$, $\Sigma X$, $PX$	the loop space, the reduced suspension, the path space
$F \to E \to B$	a fibration with fibre $F$, total space $E$ and base $B$
$h_n : \pi_n \to H_n$	the Hurewicz homomorphism
$D^n$, $S^n$, $X^n$	the disc, the sphere and the $n$-skeleton
$\mathcal{W}$, $\mathcal{C}\mathrm{of}$, $\mathcal{F}\mathrm{ib}$	weak equivalences, cofibrations, fibrations
$\operatorname{Ho}(\mathcal{C})$, $QX$, $RX$	the homotopy category and the cofibrant and fibrant replacements
$\Sigma^\infty X$, $\mathbb{S}$, $\pi_k^s$	the suspension spectrum, the sphere spectrum, the stable stems
$K(\pi,n)$, $BG$, $EG$	Eilenberg–MacLane space, classifying space, universal bundle
$\mathcal{A}_p$, $\mathrm{Sq}^i$, $P^i$	the Steenrod algebra and its operations
$\mathcal{SH}$	the stable homotopy category