This is a glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, timeline of manifolds.
Convention: Throughout the article, I denotes the unit interval, Sn the n-sphere and Dn the n-disk. Also, throughout the article, spaces are assumed to be reasonable; this can be taken to mean for example, a space is a CW complex or compactly generated weakly Hausdorff space. Similarly, no attempt is made to be definitive about the definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations. Inclusion criterion: As there is no glossary of homological algebra in Wikipedia right now, this glossary also includes a few concepts in homological algebra (e.g., chain homotopy); some concepts in geometric topology and differential topology are also fair game. On the other hand, the items that appear in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory covers (or will cover) concepts in theory of model categories. See the glossary of symplectic geometry for the topics in symplectic topology such as quantization.
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* The base point of a based space.
X + {\displaystyle X_{+}}
For an unbased space X, X+ is the based space obtained by adjoining a disjoint base point.
A
absolute neighborhood retract An absolute neighborhood retract is used as an alternative to a CW-complex as a nice space (the homotopy type of an ANR and that of a CW-complex are the same).
abstract 1. Abstract homotopy theory
Adams 1. John Frank Adams. 2. The Adams spectral sequence. 3. The Adams conjecture. 4. The Adams e-invariant. 5. The Adams operations.
Alexander duality Alexander duality
Alexander trick The Alexander trick produces a section of the restriction map Top ( D n + 1 ) → Top ( S n ) {\displaystyle \operatorname {Top} (D^{n+1})\to \operatorname {Top} (S^{n})} , Top denoting a homeomorphism group; namely, the section is given by sending a homeomorphism f : S n → S n {\displaystyle f:S^{n}\to S^{n}} to the homeomorphism
f ~ : D n + 1 → D n + 1 , 0 ↦ 0 , 0 ≠ x ↦ | x | f ( x / | x | ) {\displaystyle {\widetilde {f}}:D^{n+1}\to D^{n+1},\,0\mapsto 0,0\neq x\mapsto |x|f(x/|x|)} . This section is in fact a homotopy inverse.
Analysis Situs Analysis Situs.
annulus The annulus theorem.
approximate fibration 1. An approximate fibration, a generalization of a fibration and a projection in a locally trivial bundle. 2. A manifold approximate fibration is a proper approximate fibration between manifolds.
aspherical space Aspherical space
assembly map
Atiyah 1. Michael Atiyah. 2. Atiyah duality. 3. The Atiyah–Hirzebruch spectral sequence.
B
bar construction
based space A pair (X, x0) consisting of a space X and a point x0 in X.
Betti number See Betti number. Bing–Borsuk conjecture See Bing–Borsuk conjecture.
Bockstein homomorphism
Borel Borel conjecture.
Borel–Moore homology
Borsuk's theorem
Bott 1. Raoul Bott. 2. The Bott periodicity theorem for unitary groups say: π q U = π q + 2 U , q ≥ 0 {\displaystyle \pi _{q}U=\pi _{q+2}U,q\geq 0} . 3. The Bott periodicity theorem for orthogonal groups say: π q O = π q + 8 O , q ≥ 0 {\displaystyle \pi _{q}O=\pi _{q+8}O,q\geq 0} .
Brouwer fixed-point theorem The Brouwer fixed-point theorem says that any map f : D n → D n {\displaystyle f:D^{n}\to D^{n}} has a fixed point.
Brown–Peterson spectrum The Brown–Peterson spectrum.
C
cap product
Casson Casson invariant.
Čech cohomology
cellular 1. A map ƒ:X→Y between CW complexes is cellular if f ( X n ) ⊂ Y n {\displaystyle f(X^{n})\subset Y^{n}} for all n. 2. The cellular approximation theorem says that every map between CW complexes is homotopic to a cellular map between them. 3. The cellular homology is the (canonical) homology of a CW complex. Note it applies to CW complexes and not to spaces in general. A cellular homology is highly computable; it is especially useful for spaces with natural cell decompositions like projective spaces or Grassmannian.
chain homotopy Given chain maps f , g : ( C , d C ) → ( D , d D ) {\displaystyle f,g:(C,d_{C})\to (D,d_{D})} between chain complexes of modules, a chain homotopy s from f to g is a sequence of module homomorphisms s i : C i → D i + 1 {\displaystyle s_{i}:C_{i}\to D_{i+1}} satisfying f i − g i = d D ∘ s i + s i − 1 ∘ d C {\displaystyle f_{i}-g_{i}=d_{D}\circ s_{i}+s_{i-1}\circ d_{C}} . It is also called a homotopy operator.
chain map A chain map f : ( C , d C ) → ( D , d D ) {\displaystyle f:(C,d_{C})\to (D,d_{D})} between chain complexes of modules is a sequence of module homomorphisms f i : C i → D i {\displaystyle f_{i}:C_{i}\to D_{i}} that commutes with the differentials; i.e., d D ∘ f i = f i − 1 ∘ d C {\displaystyle d_{D}\circ f_{i}=f_{i-1}\circ d_{C}} .
chain homotopy equivalence A chain map that is an isomorphism up to chain homotopy; that is, if ƒ:C→D is a chain map, then it is a chain homotopy equivalence if there is a chain map g:D→C such that gƒ and ƒg are chain homotopic to the identity homomorphisms on C and D, respectively.
change of fiber The change of fiber of a fibration p is a homotopy equivalence, up to homotopy, between the fibers of p induced by a path in the base.
character variety The character variety of a group π and an algebraic group G (e.g., a reductive complex Lie group) is the geometric invariant theory quotient by G:
X ( π , G ) = Hom ( π , G ) / / G {\displaystyle {\mathcal {X}}(\pi ,G)=\operatorname {Hom} (\pi ,G)/\!/G} .
characteristic class Let Vect(X) be the set of isomorphism classes of vector bundles on X. We can view X ↦ Vect ( X ) {\displaystyle X\mapsto \operatorname {Vect} (X)} as a contravariant functor from Top to Set by sending a map ƒ:X → Y to the pullback ƒ* along it. Then a characteristic class is a natural transformation from Vect to the cohomology functor H*. Explicitly, to each vector bundle E we assign a cohomology class, say, c(E). The assignment is natural in the sense that ƒ*c(E) = c(ƒ*E).
chromatic homotopy theory chromatic homotopy theory.
class 1. Chern class. 2. Stiefel–Whitney class.
classifying space Loosely speaking, a classifying space is a space representing some contravariant functor defined on the category of spaces; for example, B U {\displaystyle BU} is the classifying space in the sense [ − , B U ] {\displaystyle [-,BU]} is the functor X ↦ Vect R ( X ) {\displaystyle X\mapsto \operatorname {Vect} ^{\mathbb {R} }(X)} that sends a space to the set of isomorphism classes of real vector bundles on the space.
clutching
cobar spectral sequence
cobordism 1. See cobordism. 2. A cobordism ring is a ring whose elements are cobordism classes. 3. See also h-cobordism theorem, s-cobordism theorem.
coefficient ring If E is a ring spectrum, then the coefficient ring of it is the ring π ∗ E {\displaystyle \pi _{*}E} .
cofiber sequence A cofiber sequence is any sequence that is equivalent to the sequence X → f Y → C f {\displaystyle X{\overset {f}{\to }}Y\to C_{f}} for some ƒ where C f {\displaystyle C_{f}} is the reduced mapping cone of ƒ (called the cofiber of ƒ).
cofibrant approximation
cofibration A map i : A → B {\displaystyle i:A\to B} is a cofibration if it satisfies the property: given h 0 : B → X {\displaystyle h_{0}:B\to X} and homotopy g t : A → X {\displaystyle g_{t}:A\to X} such that g 0 = h 0 ∘ i {\displaystyle g_{0}=h_{0}\circ i} , there is a homotopy h t : B → X {\displaystyle h_{t}:B\to X} such that h t ∘ i = g t {\displaystyle h_{t}\circ i=g_{t}} . A cofibration is injective and is a homeomorphism onto its image.
coherent homotopy
coherency See coherency (homotopy theory)
cohomotopy group For a based space X, the set of homotopy classes [ X , S n ] {\displaystyle [X,S^{n}]} is called the n-th cohomotopy group of X.
cohomology operation
collapse An informal phrase but usually means taking a quotient; e.g., a cone is obtained by collapsing the top (or bottom) of a cylinder.
collar collar neighbourhood
completion
complex bordism
complex-oriented A multiplicative cohomology theory E is complex-oriented if the restriction map E2(CP∞) → E2(CP1) is surjective.
concordant
cone The cone over a space X is C X = X × I / X × { 0 } {\displaystyle CX=X\times I/X\times \{0\}} . The reduced cone is obtained from the reduced cylinder X ∧ I + {\displaystyle X\wedge I_{+}} by collapsing the top.
connective A spectrum E is connective if π q E = 0 {\displaystyle \pi _{q}E=0} for all negative integers q.
configuration space
constant A constant sheaf on a space X is a sheaf F {\displaystyle {\mathcal {F}}} on X such that for some set A and some map A → F ( X ) {\displaystyle A\to {\mathcal {F}}(X)} , the natural map A → F ( X ) → F x {\displaystyle A\to {\mathcal {F}}(X)\to {\mathcal {F}}_{x}} is bijective for any x in X.
continuous Continuous cohomology.
contractible space A space is contractible if the identity map on the space is homotopic to the constant map.
covering 1. A map p: Y → X is a covering or a covering map if each point of x has a neighborhood N that is evenly covered by p; this means that the pre-image of N is a disjoint union of open sets, each of which maps to N homeomorphically. 2. It is n-sheeted if each fiber p−1(x) has exactly n elements. 3. It is universal if Y is simply connected. 4. A morphism of a covering is a map over X. In particular, an automorphism of a covering p:Y→X (also called a deck transformation) is a map Y→Y over X that has inverse; i.e., a homeomorphism over X. 5. A G-covering is a covering arising from a group action on a space X by a group G, the covering map being the quotient map from X to the orbit space X/G. The notion is used to state the universal property: if X admits a universal covering (in particular connected), then
Hom ( π 1 ( X , x 0 ) , G ) {\displaystyle \operatorname {Hom} (\pi _{1}(X,x_{0}),G)} is the set of isomorphism classes of G-coverings. In particular, if G is abelian, then the left-hand side is Hom ( π 1 ( X , x 0 ) , G ) = H 1 ( X ; G ) {\displaystyle \operatorname {Hom} (\pi _{1}(X,x_{0}),G)=\operatorname {H} ^{1}(X;G)} (cf. nonabelian cohomology.) 6. covering dimension.
cup product
CW complex A CW complex is a space X equipped with a CW structure; i.e., a filtration
X 0 ⊂ X 1 ⊂ X 2 ⊂ ⋯ ⊂ X {\displaystyle X^{0}\subset X^{1}\subset X^{2}\subset \cdots \subset X}
such that (1) X0 is discrete and (2) Xn is obtained from Xn-1 by attaching n-cells.
cyclic homology
D
deck transformation Another term for an automorphism of a covering.
deformation retract A subspace A ⊂ X {\displaystyle A\subset X} is called a deformation retract of X if there is a homotopy h t : X → X {\displaystyle h_{t}:X\to X} such that h 0 {\displaystyle h_{0}} is the identity, h 1 ( X ) ⊂ A {\displaystyle h_{1}(X)\subset A} and h 1 | A {\displaystyle {h_{1}}|_{A}} is the identity (i.e., h 1 {\displaystyle h_{1}} is a retract of A ↪ X {\displaystyle A\hookrightarrow X} in the sense in category theory). It is called a strong deformation retract if, in addition, h t {\displaystyle h_{t}} satisfies the requirement that h t | A {\displaystyle {h_{t}}|_{A}} is the identity. For example, a homotopy h t : B → B , x ↦ ( 1 − t ) x {\displaystyle h_{t}:B\to B,\,x\mapsto (1-t)x} exhibits that the origin is a strong deformation retract of an open ball B centered at the origin.
Deligne–Beilinson cohomology Deligne–Beilinson cohomology
delooping
degeneracy cycle
degree
de Rham 1. de Rham cohomology, the cohomology of complex of differential forms. 2. The de Rham theorem gives an explicit isomorphism between the de Rham cohomology and the singular cohomology.
disjoint disk property disjoint disk property.
Dold The Dold–Thom theorem.
dominate A space Y {\displaystyle Y} is said to dominate a space X {\displaystyle X} if there are p : Y → X {\displaystyle p:Y\to X} and g : X → Y {\displaystyle g:X\to Y} such that p ∘ g : X → X {\displaystyle p\circ g:X\to X} is homotopic to the identity.
E
Eckmann–Hilton argument The Eckmann–Hilton argument.
Eckmann–Hilton duality
Eilenberg–MacLane spaces Given an abelian group π, the Eilenberg–MacLane spaces K ( π , n ) {\displaystyle K(\pi ,n)} are characterized by
π q K ( π , n ) = { π if q = n 0 otherwise {\displaystyle \pi _{q}K(\pi ,n)={\begin{cases}\pi &{\text{if }}q=n\\0&{\text{otherwise}}\end{cases}}} .
Eilenberg–Steenrod axioms The Eilenberg–Steenrod axioms are the set of axioms that any cohomology theory (singular, cellular, etc.) must satisfy. Weakening the axioms (namely dropping the dimension axiom) leads to a generalized cohomology theory.
Eilenberg–Zilber theorem
elliptic elliptic cohomology.
En-algebra
equivariant algebraic topology Equivariant algebraic topoloy is the study of spaces with (continuous) group action.
etale étale homotopy.
Euclidean A Euclidean neighborhood retract
exact A sequence of pointed sets X → f Y → g Z {\displaystyle X{\overset {f}{\to }}Y{\overset {g}{\to }}Z} is exact if the image of f coincides with the pre-image of the chosen point of Z.
excision The excision axiom for homology says: if U ⊂ X {\displaystyle U\subset X} and U ¯ ⊂ int ( A ) {\displaystyle {\overline {U}}\subset \operatorname {int} (A)} , then for each q,
H q ( X − U , A − U ) → H q ( X , A ) {\displaystyle \operatorname {H} _{q}(X-U,A-U)\to \operatorname {H} _{q}(X,A)}
is an isomorphism.
excisive pair/triad
F
factorization homology
fiber-homotopy equivalence Given D→B, E→B, a map ƒ:D→E over B is a fiber-homotopy equivalence if it is invertible up to homotopy over B. The basic fact is that if D→B, E→B are fibrations, then a homotopy equivalence from D to E is a fiber-homotopy equivalence.
fiber sequence The fiber sequence of a map f : X → Y {\displaystyle f:X\to Y} is the sequence F f → p X → f Y {\displaystyle F_{f}{\overset {p}{\to }}X{\overset {f}{\to }}Y} where F f → p X {\displaystyle F_{f}{\overset {p}{\to }}X} is the homotopy fiber of f; i.e., the pullback of the path space fibration P Y → Y {\displaystyle PY\to Y} along f.
fiber square fiber square
fibration A map p:E → B is a fibration if for any given homotopy g t : X → B {\displaystyle g_{t}:X\to B} and a map h 0 : X → E {\displaystyle h_{0}:X\to E} such that p ∘ h 0 = g 0 {\displaystyle p\circ h_{0}=g_{0}} , there exists a homotopy h t : X → E {\displaystyle h_{t}:X\to E} such that p ∘ h t = g t {\displaystyle p\circ h_{t}=g_{t}} . (The above property is called the homotopy lifting property.) A covering map is a basic example of a fibration.
fibration sequence One says F → X → p B {\displaystyle F\to X{\overset {p}{\to }}B} is a fibration sequence to mean that p is a fibration and that F is homotopy equivalent to the homotopy fiber of p, with some understanding of base points.
finitely dominated
fundamental class
fundamental group The fundamental group of a space X with base point x0 is the group of homotopy classes of loops at x0. It is precisely the first homotopy group of (X, x0) and is thus denoted by π 1 ( X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} .
fundamental groupoid The fundamental groupoid of a space X is the category whose objects are the points of X and whose morphisms x → y are the homotopy classes of paths from x to y; thus, the set of all morphisms from an object x0 to itself is, by definition, the fundamental group π 1 ( X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} .
framed A framed manifold is a manifold with a framing.
free Synonymous with unbased. For example, the free path space of a space X refers to the space of all maps from I to X; i.e., X I {\displaystyle X^{I}} while the path space of a based space X consists of such map that preserve the base point (i.e., 0 goes to the base point of X).
Freedman Freedman's E8 manifold.
Freudenthal suspension theorem For a nondegenerately based space X, the Freudenthal suspension theorem says: if X is (n-1)-connected, then the suspension homomorphism
π q X → π q + 1 Σ X {\displaystyle \pi _{q}X\to \pi _{q+1}\Sigma X}
is bijective for q < 2n - 1 and is surjective if q = 2n - 1.
Fulton–MacPherson compactification The Fulton–MacPherson compactification of the configuration space of n distinct labeled points in a compact complex manifold is a natural smooth compactification introduced by Fulton and MacPherson.
G
G-fibration A G-fibration with some topological monoid G. An example is Moore's path space fibration.
G-space A G-space is a space together with an action of a group G (usually satisfying some conditions).
Γ-space
generalized cohomology theory A generalized cohomology theory is a contravariant functor from the category of pairs of spaces to the category of abelian groups that satisfies all of the Eilenberg–Steenrod axioms except the dimension axiom.
geometrization conjecture geometrization conjecture
genus
germ germ
group completion grouplike An H-space X is said to be group-like or grouplike if π 0 X {\displaystyle \pi _{0}X} is a group; i.e., X satisfies the group axioms up to homotopy.
Gysin sequence
H
Hauptvermutung 1. Hauptvermutung, a German for main conjecture, is short for die Hauptvermutung der kombinatorischen Topologie (the main conjecture of combinatorial topology). It asks whether two simplicial complexes are isomorphic if homeomorphic. It was disproved by Milnor in 1961. 2. There are some variants; for example, one can ask whether two PL manifolds are PL-isomorphic if homeomorphic (which is also false).
h-cobordism h-cobordism.
Hilton–Milnor theorem The Hilton–Milnor theorem.
Hirzebruch Hirzebruch signature theorem.
H-space An H-space is a based space that is a unital magma up to homotopy.
Hodge The Hodge spectral sequence.
homologous Two cycles are homologous if they belong to the same homology class.
homology manifold A homology manifold is a space that looks like a topological manifold, homology-theory speaking.
homology sphere A homology sphere is a manifold having the homology type of a sphere.
homotopy category Let C be a subcategory of the category of all spaces. Then the homotopy category of C is the category whose class of objects is the same as the class of objects of C but the set of morphisms from an object x to an object y is the set of the homotopy classes of morphisms from x to y in C. For example, a map is a homotopy equivalence if and only if it is an isomorphism in the homotopy category.
homotopy colimit A homotopy colimit is a homotopically-correct version of colimit.
homotopy over a space B A homotopy ht such that for each fixed t, ht is a map over B.
homotopy equivalence 1. A map ƒ:X→Y is a homotopy equivalence if it is invertible up to homotopy; that is, there exists a map g: Y→X such that g ∘ ƒ is homotopic to th identity map on X and ƒ ∘ g is homotopic to the identity map on Y. 2. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between the two. For example, by definition, a space is contractible if it is homotopy equivalent to a point space.
homotopy excision theorem The homotopy excision theorem is a substitute for the failure of excision for homotopy groups.
homotopy fiber The homotopy fiber of a based map ƒ:X→Y, denoted by Fƒ, is the pullback of P Y → Y , χ ↦ χ ( 1 ) {\displaystyle PY\to Y,\,\chi \mapsto \chi (1)} along f.
homotopy fiber product A fiber product is a particular kind of a limit. Replacing this limit lim with a homotopy limit holim yields a homotopy fiber product.
homotopy group 1. For a based space X, let π n X = [ S n , X ] {\displaystyle \pi _{n}X=[S^{n},X]} , the set of homotopy classes of based maps. Then π 0 X {\displaystyle \pi _{0}X} is the set of path-connected components of X, π 1 X {\displaystyle \pi _{1}X} is the fundamental group of X and π n X , n ≥ 2 {\displaystyle \pi _{n}X,\,n\geq 2} are the (higher) n-th homotopy groups of X. 2. For based spaces A ⊂ X {\displaystyle A\subset X} , the relative homotopy group π n ( X , A ) {\displaystyle \pi _{n}(X,A)} is defined as π n − 1 {\displaystyle \pi _{n-1}} of the space of paths that all start at the base point of X and end somewhere in A. Equivalently, it is the π n − 1 {\displaystyle \pi _{n-1}} of the homotopy fiber of A ↪ X {\displaystyle A\hookrightarrow X} . 3. If E is a spectrum, then π k E = lim → n π k + n E n . {\displaystyle \pi _{k}E=\varinjlim _{n}\pi _{k+n}E_{n}.}
4. If X is a based space, then the stable k-th homotopy group of X is π k s X = lim → n π k + n Σ n X {\displaystyle \pi _{k}^{s}X=\varinjlim _{n}\pi _{k+n}\Sigma ^{n}X} . In other words, it is the k-th homotopy group of the suspension spectrum of X.
homotopy pullback A homotopy pullback is a special case of a homotopy limit that is a homotopically-correct pullback.
homotopy quotient If G is a Lie group acting on a manifold X, then the quotient space ( E G × X ) / G {\displaystyle (EG\times X)/G} is called the homotopy quotient (or Borel construction) of X by G, where EG is the universal bundle of G.
homotopy spectral sequence
homotopy sphere A homotopy sphere is a manifold having the homotopy type of a sphere.
Hopf 1. Heinz Hopf. 2. Hopf invariant. 3. The Hopf index theorem. 4. Hopf construction.
Hurewicz The Hurewicz theorem establishes a relationship between homotopy groups and homology groups.
I
infinite loop space
infinite loop space machine Infinite loop space machine.
infinite mapping telescope
intersection intersection pairing. intersection homology, a substitute for an ordinary (singular) homology for a singular space. intersection cohomology
integration along the fiber See integration along the fiber.
invariance of domain invariance of domain.
isotopy
J
J-homomorphism See J-homomorphism.
join The join of based spaces X, Y is X ⋆ Y = Σ ( X ∧ Y ) . {\displaystyle X\star Y=\Sigma (X\wedge Y).}
K
k-invariant
Kan complex See Kan complex.
Kirby–Siebenmann Kirby–Siebenmann classification.
Kervaire invariant The Kervaire invariant.
Koszul duality Koszul duality.
Kuiper Kuiper's theorem says that the general linear group of an infinite-dimensional Hilbert space is contractible.
Künneth formula
L
L-class L-class.
Lazard ring The Lazard ring L is the (huge) commutative ring together with the formal group law ƒ that is universal among all the formal group laws in the sense that any formal group law g over a commutative ring R is obtained via a ring homomorphism L → R mapping ƒ to g. According to Quillen's theorem, it is also the coefficient ring of the complex bordism MU. The Spec of L is called the moduli space of formal group laws.
Lefschetz 1. Solomon Lefschetz 2. The Lefschetz fixed-point theorem says: given a finite simplicial complex K and its geometric realization X, if a map f : X → X {\displaystyle f:X\to X} has no fixed point, then the Lefschetz number of f; that is,
∑ 0 ∞ ( − 1 ) q tr ( f ∗ : H q ( X ) → H q ( X ) ) {\displaystyle \sum _{0}^{\infty }(-1)^{q}\operatorname {tr} (f_{*}:\operatorname {H} _{q}(X)\to \operatorname {H} _{q}(X))}
is zero. For example, it implies the Brouwer fixed-point theorem since the Lefschetz number of f : D n → D n {\displaystyle f:D^{n}\to D^{n}} is, as higher homologies vanish, one. 3. The Lefschetz hyperplane theorem.
lens space The lens space is the quotient space { z ∈ C n | | z | = 1 } / μ p {\displaystyle \{z\in \mathbb {C} ^{n}||z|=1\}/\mu _{p}} where μ p {\displaystyle \mu _{p}} is the group of p-th r