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 generatedweakly 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.
The Alexander trick produces a section of the restriction map , Top denoting a homeomorphism group; namely, the section is given by sending a homeomorphism to the homeomorphism
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 between chain complexes of modules, a chain homotopys from f to g is a sequence of module homomorphisms satisfying . It is also called a homotopy operator.
chain map
A chain map between chain complexes of modules is a sequence of module homomorphisms that commutes with the differentials; i.e., .
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.
Let Vect(X) be the set of isomorphism classes of vector bundles on X. We can view 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).
Loosely speaking, a classifying space is a space representing some contravariant functor defined on the category of spaces; for example, is the classifying space in the sense is the functor that sends a space to the set of isomorphism classes of real vector bundles on the space.
If E is a ring spectrum, then the coefficient ring of it is the ring .
cofiber sequence
A cofiber sequence is any sequence that is equivalent to the sequence for some ƒ where is the reduced mapping cone of ƒ (called the cofiber of ƒ).
cofibrant approximation
cofibration
A map is a cofibration if it satisfies the property: given and homotopy such that , there is a homotopy such that .[3] A cofibration is injective and is a homeomorphism onto its image.
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.
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 spaceX/G. The notion is used to state the universal property: if X admits a universal covering (in particular connected), then
is the set of isomorphism classes of G-coverings.
In particular, if G is abelian, then the left-hand side is (cf. nonabelian cohomology.)
cup product
CW complex
A CW complex is a space X equipped with a CW structure; i.e., a filtration
such that (1) X0 is discrete and (2) Xn is obtained from Xn-1 by attaching n-cells.
A subspace is called a deformation retract of X if there is a homotopy such that is the identity, and is the identity (i.e., is a retract of in the sense in category theory). It is called a strong deformation retract if, in addition, satisfies the requirement that is the identity. For example, a homotopy exhibits that the origin is a strong deformation retract of an open ball B centered at the origin.
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.
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 is the sequence where is the homotopy fiber of f; i.e., the pullback of the path space fibration along f.
A map p:E → B is a fibration if for any given homotopy and a map such that , there exists a homotopy such that . (The above property is called the homotopy lifting property.) A covering map is a basic example of a fibration.
fibration sequence
One says 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 .
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 fundament group .
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., 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).
Freudenthal suspension theorem
For a nondegenerately based space X, the Freudenthal suspension theorem says: if X is (n-1)-connected, then the suspension homomorphism
is bijective for q < 2n - 1 and is surjective if q = 2n - 1.
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.
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).
Two cycles are homologus if they belong to the same homology class.
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.
1. For a based space X, let , the set of homotopy classes of based maps. Then is the set of path-connected components of X, is the fundamental group of X and are the (higher) n-th homotopy groups of X.
2. For based spaces , the relative homotopy group is defined as of the space of paths that all start at the base point of X and end somewhere in A. Equivalently, it is the of the homotopy fiber of .
3. If E is a spectrum, then
4. If X is a based space, then the stable k-th homotopy group of X is . 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 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.
The Lazard ringL 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.
2. The Lefschetz fixed-point theorem says: given a finite simplicial complex K and its geometric realization X, if a map has no fixed point, then the Lefschetz number of f; that is,
is zero. For example, it implies the Brouwer fixed-point theorem since the Lefschetz number of is, as higher homologies vanish, one.
1. A module over the group ring for some based space B; in other words, an abelian group together with a homomorphism .
2. The local coefficient system over a based space B with an abelian group A is a fiber bundle over B with discrete fiber A. If B admits a universal covering , then this meaning coincides with that of 1. in the sense: every local coefficient system over B can be given as the associated bundle.
3. The reduced versions of the above are obtained by using reduced cone and reduced cylinder.
4. The mapping path spacePp of a map p:E→B is the pullback of along p. If p is fibration, then the natural map E→Pp is a fiber-homotopy equivalence; thus, one can replace E by the mapping path space without changing the homotopy type of the fiber. A mapping path space is also called a mapping cocylinder.
5. As a set, the mapping space from a space X to a space Y is the set of all continuous maps from X to Y. It is topologized in such a way the mapping space is a space; tha is, an object in the category of spaces used in algebraic topology; e.g., the category of compactly generated weak Hausdorff spaces. This topology may or may not be compact-open topology.
A generalized cohomology theoryE is multiplicative if E*(X) is a graded ring. For example, the ordinary cohomology theory and the complex K-theory are multiplicative (in fact, cohomology theories defined by E∞-rings are multiplicative.)
A based space X is n-connected if for all integers q ≤ n. For example, "1-connected" is the same thing as "simply connected".
n-equivalent
NDR-pair
A pair of spaces is said to be an NDR-pair (=neighborhood deformation retract pair) if there is a map and a homotopy such that , , and . If A is a closed subspace of X, then the pair is an NDR-pair if and only if is a cofibration.
nilpotent
1. nilpotent space; for example, a simply connected space is nilpotent.
Given a simplicial groupG, the normalized chain complexNG of G is given by with the n-th differential given by ; intuitively, one throws out degenerate chains.[5] It is also called the Moore complex.
1. The orientation covering (or orientation double cover) of a manifold is a two-sheeted covering so that each fiber over x corresponds to two different ways of orienting a neighborhood of x.
2. An orientation of a manifold is a section of an orientation covering; i.e., a consistent choice of a point in each fiber.
An equivalence class of paths (two paths are equivalent if they are homotopic to each other).
path lifting
A path lifting function for a map p: E → B is a section of where is the mapping path space of p. For example, a covering is a fibration with a unique path lifting function. By formal consideration, a map is a fibration if and only if there is a path lifting function for it.
path space
The path space of a based space X is , the space of based maps, where the base point of I is 0. Put in another way, it is the (set-theoretic) fiber of over the base point of X. The projection is called the path space fibration, whose fiber over the base point of X is the loop space . See also mapping path space.
Not particularly a precise term. But it could mean, for example, that G is discrete and each point of the G-space has a neighborhood V such that for each g in G that is not the identity element, gV intersects V at finitely many points.
2. The rationalization of a space X is, roughly, the localization of X at zero. More precisely, X0 together with j: X → X0 is a rationalization of X if the map induced by j is an isomorphism of vector spaces and .
See simplicial complex; the basic example is a triangulation of a manifold.
simplicial homology
A simplicial homology is the (canonical) homology of a simplicial complex. Note it applies to simplicial complexes and not to spaces; cf. #singular homology.
signature invariant
singular
1. Given a space X and an abelian group π, the singular homology group of X with coefficients in π is
where is the singular chain complex of X; i.e., the n-th degree piece is the free abelian group generated by all the maps from the standard n-simplex to X. A singular homology is a special case of a simplicial homology; indeed, for each space X, there is the singular simplicial complex of X[6] whose homology is the singular homology of X.
Roughly a sequence of spaces together with the maps (called the structure maps) between the consecutive terms; see spectrum (topology).
sphere bundle
A sphere bundle is a fiber bundle whose fibers are spheres.
sphere spectrum
The sphere spectrum is a spectrum consisting of a sequence of spheres together with the maps between the spheres given by suspensions. In short, it is the suspension spectrum of .
2. If E is a vector bundle on a paracompact space X, then the Thom space of E is obtained by first replacing each fiber by its compactification and then collapsing the base X.
The van Kampen theorem says: if a space X is path-connected and if x0 is a point in X, then
where the colimit runs over some open cover of X consisting of path-connected open subsets containing x0 such that the cover is closed under finite intersections.
A map ƒ:X→Y of based spaces is a weak equivalence if for each q, the induced map is bijective.
wedge
For based spaces X, Y, the wedge product of X and Y is the coproduct of X and Y; concretely, it is obtained by taking their disjoint union and then identifying the respective base points.
well pointed
A based space is well pointed (or non-degenerately based) if the inclusion of the base point is a cofibration.
^Let r, s denote the restriction and the section. For each f in , define . Then .
^Despite the name, it may not be an algebraic variety in the strict sense; for example, it may not be irreducible. Also, without some finiteness assumption on G, it is only a scheme.
Bousfield, A. K.; Kan, D. M. (1987), Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, vol. 304, Springer, ISBN9783540061052
B. I. Dundas, M. Levine, P. A. Østvær, O. Röndigs, and V. Voevodsky. Motivic homotopy theory. Universitext. Springer-Verlag, Berlin, 2007. Lectures from the Summer School held in Nordfjordeid, August 2002. [1]