Algebraic topology

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Homotopy group

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The th homotopy group of a topological space is the set of homotopy classes of maps from the n-sphere to , with a group structure, and is denoted . The fundamental group is , and, as in the case of , the maps must pass through a basepoint . For , the homotopy group is an Abelian group.The group operations are not as simple as those for the fundamental group. Consider two maps and , which pass through . The product is given by mapping the equator to the basepoint . Then the northern hemisphere is mapped to the sphere by collapsing the equator to a point, and then it is mapped to by . The southern hemisphere is similarly mapped to by . The diagram above shows the product of two spheres.The identity element is represented by the constant map . The choice of direction of a loop in the fundamental group corresponds to a manifold orientation of in a homotopy..

Simplicial map

Let and be simplicial complexes, and let be a map. Suppose that whenever the vertices , ..., of span a simplex of , the points , ..., are vertices of a simplex of . Then can be extended to a continuous map such thatimpliesThe map is then called the linear simplicial map induced by the vertex map (Munkres 1993, p. 12).

Simplicial homomorphism

Let be a bijective correspondence such that the vertices , ..., of span a simplex of iff , ..., span a simplex of . Then the induced simplicial map is a homeomorphism, and the map is called a simplicial homeomorphism (Munkres 1993, p. 13).

Homotopy equivalence

Two topological spaces and are homotopy equivalent if there exist continuous maps and , such that the composition is homotopic to the identity on , and such that is homotopic to . Each of the maps and is called a homotopy equivalence, and is said to be a homotopy inverse to (and vice versa).One should think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another.Certainly any homeomorphism is a homotopy equivalence, with homotopy inverse , but the converse does not necessarily hold.Some spaces, such as any ball , can be deformed continuously into a point. A space with this property is said to be contractible, the precise definition being that is homotopy equivalent to a point. It is a fact that a space is contractible, if and only if the identity map is null-homotopic, i.e., homotopic to a constant map...

Cellular map

Let and be CW-complexes and let (respectively ) denote the -skeleton of (respectively ). Then a continuous map is said to be cellular if it takes -skeletons to -skeletons for all , i.e, iffor all nonnegative integers .The contents of the cellular approximation theorem is that, in a certain sense, all maps between CW-complexes can be taken to be cellular.

Homotopy class

Given two topological spaces and , place an equivalence relationship on the continuous maps using homotopies, and write if is homotopic to . Roughly speaking, two maps are homotopic if one can be deformed into the other. This equivalence relation is transitive because these homotopy deformations can be composed (i.e., one can follow the other).A simple example is the case of continuous maps from one circle to another circle. Consider the number of ways an infinitely stretchable string can be tied around a tree trunk. The string forms the first circle, and the tree trunk's surface forms the second circle. For any integer , the string can be wrapped around the tree times, for positive clockwise, and negative counterclockwise. Each integer corresponds to a homotopy class of maps from to .After the string is wrapped around the tree times, it could be deformed a little bit to get another continuous map, but it would still be in the same homotopy class,..

Cellular approximation theorem

Let and be CW-complexes, and let be a continuous map. Then the cellular approximation theorem states that any such is homotopic to a cellular map. In fact, if the map is already cellular on a CW-subcomplex of , then the homotopy can be taken to be stationary on .A famous application of the theorem is the calculation of some homotopy groups of -spheres . Indeed, let and bestow on both and their usual CW-structure, with one -cell, and one -cell, respectively one -cell. If is a continuous, base-point preserving map, then by cellular approximation, it is homotopic to a cellular map . This map must map the -skeleton of into the -skeleton of , but the -skeleton of is itself, while the -skeleton of is the zero-cell, i.e., a point. This is because of the condition . Thus is a constant map, whence ...

Simplicial complex link

The set , where is a closed star and is a star, is called the link of in a simplicial complex and is denoted (Munkres 1993, p. 11).

Bordism group

There are two types of bordism groups: bordism groups, also called cobordism groups or cobordism rings, and there are singular bordism groups. The bordism groups give a framework for getting a grip on the question, "When is a compact boundaryless manifold the boundary of another manifold?" The answer is, precisely when all its Stiefel-Whitney numbers are zero. Singular bordism groups give insight into Steenrod's realization problem: "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too.The machinery of the bordism group winds up being important for homotopytheory as well.

Simplicial complex

A simplicial complex is a space with a triangulation. Formally, a simplicial complex in is a collection of simplices in such that 1. Every face of a simplex of is in , and 2. The intersection of any two simplices of is a face of each of them (Munkres 1993, p. 7).Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes. When only simplicial complexes and simplicial subcomplexes are considered, defining homology is particularly easy (and, in fact, combinatorial because of its finite/counting nature). This kind of homology is called simplicial homology.


A continuous transformation from one function to another. A homotopy between two functions and from a space to a space is a continuous map from such that and , where denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping space from the first function to the second.Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. The concept of homotopy was first formulated by Poincaré around 1900 (Collins 2004).

Schönflies theorem

If is a simple closed curve in , the closure of one of the components of is homeomorphic with the unit 2-ball. This theorem may be proved using the Riemann mapping theorem, but the easiest proof is via Morse theory.The generalization to dimensions is called Mazur's theorem. It follows from the Schönflies theorem that any two knots of in or are equivalent.


Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps and are homotopic if there is a continuous mapsuch that and .Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane . The puncture can be thought of as an obstacle.However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces and are homotopy equivalent if there are maps and such that the composition is homotopic to the identity map of and is homotopic to the identity map of . For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible..

Algebraic topology

Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that remain invariant under both-directions continuous one-to-one (homeomorphic) transformations. The discipline of algebraic topology is popularly known as "rubber-sheet geometry" and can also be viewed as the study of disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of hole structures, and it gets the prefix "algebraic" since many hole structures are represented best by algebraic objects like groups and rings.Algebraic topology originated with combinatorial topology, but went beyond it probably for the first time in the 1930s when Čech cohomology was developed.A technical way of saying this is that algebraic topology is concerned with functors from the topological..


A subspace of is called a retract of if there is a continuous map (called a retraction) such that for all and all , 1. , and 2. . Equivalently, a subspace of is called a retract of if there is a continuous map (called a retraction) such that for all ,


Let denote a chain complex, a portion of which is shown below:Let denotes the th homology group. Then two homology cycles are said to be homologous, if their difference is a boundary, i.e., if .

Hereditarily unicoherent continuum

Let be a continuum (i.e., a compact connected metric space). Then is hereditarily unicoherent provided that every subcontinuum of is unicoherent.Any hereditarily unicoherent continuum is a unicoherent space, but there are unicoherent continua that are not hereditarily unicoherent. For example, the unit interval is hereditarily unicoherent, but a ray winding down on a circle is not hereditarily unicoherent, even though it is unicoherent. (This is due to the fact that a circle is not unicoherent.)

Absolute retract

Let be a class of topological spaces that is closed under homeomorphism, and let be a topological space. If and for every such that , is a retract of , then is an absolute retract for the class .These notions can be generalized to category theory, and because there are category-theoretic versions, there are also other more specific versions, as in universal algebra and modern algebra.

Wild point

For any point on the boundary of an ordinary ball, find a neighborhood of in which the intersection with the ball's boundary cuts the neighborhood into two parts, each homeomorphic to a ball. A wild point is a point on the boundary that has no such neighborhood.

Chern class

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The th Chern class is an obstruction to the existence of everywhere complex linearly independent vector fields on that vector bundle. The th Chern class is in the th cohomology group of the base space.

Homotopy sphere

An -dimensional manifold is said to be a homotopy sphere, if it is homotopy equivalent to the -sphere . Thus no homotopy group can distinguish between and .The solution of the generalized Poincaré conjecture in the positive implies that any compact homotopy sphere is in fact homeomorphic to a sphere.

Characteristic class

Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the existence of everywhere linearly independent vector fields on the vector bundle. The most common examples of characteristic classes are the Chern, Pontryagin, and Stiefel-Whitney classes.

Homotopy axiom

One of the Eilenberg-Steenrod axioms which states that, if is homotopic to , then their induced maps and are the same.

Topological entropy

The topological entropy of a map is defined aswhere is a partition of a bounded region containing a probability measure which is invariant under , and sup is the supremum.

Lifting problem

Given a map from a space to a space and another map from a space to a space , does there exist a map from to such that ? If such a map exists, then is called a lift of .


Given a map from a space to a space and another map from a space to a space , a lift is a map from to such that . In other words, a lift of is a map such that the diagram (shown below) commutes. If is the identity from to , a manifold, and if is the bundle projection from the tangent bundle to , the lifts are precisely vector fields. If is a bundle projection from any fiber bundle to , then lifts are precisely sections. If is the identity from to , a manifold, and a projection from the orientation double cover of , then lifts exist iff is an orientable manifold.If is a map from a circle to , an -manifold, and the bundle projection from the fiber bundle of alternating n-forms on , then lifts always exist iff is orientable. If is a map from a region in the complex plane to the complex plane (complex analytic), and if is the exponential map, lifts of are precisely logarithms of ...

Deformation retract

A subspace of is called a deformation retract of if there is a homotopy (called a retract) such that for all and , 1. , 2. , and 3. . A tightening of the last condition gives a so-called strongdeformation retract (Bredon 1993, pp. 45-46).Note that a deformation retract is also a retract, because the homotopy defines a continuous map

Suslin's theorem

A set in a Polish space is a Borel set iff it is both analytic and coanalytic. For subsets of , a set is iff it is "hyperarithmetic."

Structurally stable

A map where is a manifold is structurally stable if any perturbation is topologically conjugate to . Here, perturbation means a function such that is close to and the first derivatives of are close to those of .

Strong deformation retract

A subspace of is called a strong deformation retract of if there is a homotopy (called a retract) such that for all , , and , 1. , 2. , and 3. . If the last equation is required only for , the retract is called simply a deformation retract.

Jordan curve theorem

If is a simple closed curve in , then the Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966) states that has two components (an "inside" and "outside"), with the boundary of each.The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher (2002, p. 169), or in classic texts such as Spanier (1966). Recently, a proof checker was used by a Japanese-Polish team to create a "computer-checked" proof of the theorem (Grabowski 2005).

Cup product

The cup product is a product on cohomology classes. In the case of de Rham cohomology, a cohomology class can be represented by a closed form. The cup product of and is represented by the closed form , where is the wedge product of differential forms. It is the dual operation to intersection in homology.In general, the cup product is a mapwhich satisfies , where is the th cohomology group.


A special nonsingular map from one manifold to another such that at every point in the domain of the map, the derivative is an injective linear transformation. This is equivalent to saying that every point in the domain has a neighborhood such that, up to diffeomorphisms of the tangent space, the map looks like the inclusion map from a lower-dimensional Euclidean space to a higher-dimensional Euclidean space.

Hopf map

The first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic. Its discovery was a shock to the mathematical community, since it was believed at the time that all such maps were null-homotopic, by analogy with homology groups.The Hopf map arises in many contexts, and can be generalized to a map . For any point in the sphere, its preimage is a circle in . There are several descriptions of the Hopf map, also called the Hopf fibration.As a submanifold of , the 3-sphere is(1)and the 2-sphere is a submanifold of ,(2)The Hopf map takes points (, , , ) on a 3-sphere to points on a 2-sphere (, , )(3)(4)(5)Every point on the 2-sphere corresponds to a circlecalled the Hopf circle on the 3-sphere.By stereographic projection, the 3-sphere can be mapped to , where the point at infinity corresponds to the north pole. As a map, from , the Hopf map can be pretty complicated. The diagram above shows some of..

Hopf invariant one theorem

The Hopf invariant one theorem, sometimes also called Adams' theorem, is a deep theorem in homotopy theory which states that the only -spheres which are H-spaces are , , , and . The theorem was proved by Adams (1958, 1960).

Closed star

The set closure of a star at a vertex of a simplicial complex .

Homotopy type

A class formed by sets in which have essentially the same structure, regardless of size, shape and dimension. The "essential structure" is what a set keeps when it is transformed by compressing or dilating its parts, but without cutting or gluing. The most important feature that is preserved is the system of internal closed paths. In particular, the fundamental group remains unchanged. This object, however, only characterizes the loops, i.e., the paths which are essentially circular lines, whereas the homotopy type also refers to higher dimensional closed paths, which correspond to the boundaries of -spheres. Hence the homotopy type yields a more precise classification of geometric objects. As for the circular paths, it makes no difference whether the object is located in the plane or on the surface of a sphere, so the fundamental group is the same in both cases.The homotopy type, however, is different, since the plane does not..

Chern number

The Chern number is defined in terms of the Chern class of a manifold as follows. For any collection Chern classes such that their cup product has the same dimension as the manifold, this cup product can be evaluated on the manifold's fundamental class. The resulting number is called the Chern number for that combination of Chern classes. The most important aspect of Chern numbers is that they are bordism invariant.


In algebraic topology, a -skeleton is a simplicial subcomplex of that is the collection of all simplices of of dimension at most , denoted .The graph obtained by replacing the faces of a polyhedron with its edges and vertices is therefore the skeleton of the polyhedron. The polyhedral graphs corresponding to the skeletons of Platonic solids are illustrated above. The number of topologically distinct skeletons with graph vertices for , 5, 6, ... are 1, 2, 7, 18, 52, ... (OEIS A006869).

Deck transformation

Deck transformations, also called covering transformations, are defined for any cover . They act on by homeomorphisms which preserve the projection . Deck transformations can be defined by lifting paths from a space to its universal cover , which is a simply connected space and is a cover of . Every loop in , say a function on the unit interval with , lifts to a path , which only depends on the choice of , i.e., the starting point in the preimage of . Moreover, the endpoint depends only on the homotopy class of and . Given a point , and , a member of the fundamental group of , a point is defined to be the endpoint of a lift of a path which represents .The deck transformations of a universal cover form a group , which is the fundamental group of the quotient spaceFor example, when is the square torus then is the plane and the preimage is a translation of the integer lattice . Any loop in the torus lifts to a path in the plane, with the endpoints lying in the integer lattice...

Morse theory

A generalization of calculus of variations which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology of the manifold. For example, if a compact manifold admits a function whose only stationary points are a maximum and a minimum, then the manifold is a sphere. Technically speaking, Morse theory applied to a function on a manifold with and shows that every bordism can be realized as a finite sequence of surgeries. Conversely, a sequence of surgeries gives a bordism.There are a number of classical applications of Morse theory, including counting geodesics on a Riemann surface and determination of the topology of a Lie group (Bott 1960, Milnor 1963). Morse theory has received much attention in the last two decades as a result of the paper by Witten (1982) which relates Morse theory to quantum field theory and also directly connects the stationary points of a smooth function to..

Commutative diagram

A commutative diagram is a collection of maps in which all map compositions starting from the same set and ending with the same set give the same result. In symbols this means that, whenever one can form two sequences(1)and(2)the following equality holds:(3)Commutative diagrams are usually composed by commutative triangles and commutative squares.Commutative triangles and squares can also be combined to form plane figures or space arrangements.A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.A looped arrow indicates a map from a set to itself.The above commutative diagram expresses the fact that is the inverse map to , since it is a pictorial translation of the map equalities and .This can also be represented using two separate diagrams.Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated..

Pontryagin number

The Pontryagin number is defined in terms of the Pontryagin class of a manifold as follows. For any collection of Pontryagin classes such that their cup product has the same dimension as the manifold, this cup product can be evaluated on the manifold's fundamental class. The resulting number is called the Pontryagin number for that combination of Pontryagin classes. The most important aspect of Pontryagin numbers is that they are bordism invariant. Together, Pontryagin and Stiefel-Whitney numbers determine an oriented manifold's oriented bordism class.

Pontryagin duality

Let be a locally compact Abelian group. Let be the group of all continuous homeomorphisms , in the compact open topology. Then is also a locally compact Abelian group, where the asterisk defines a contravariant equivalence of the category of locally compact Abelian groups with itself. The natural mapping , sending to , where , is an isomorphism and a homeomorphism. Under this equivalence, compact groups are sent to discrete groups and vice versa.

Geometric realization

If the abstract simplicial complex is isomorphic with the vertex scheme of the simplicial complex , then is said to be a geometric realization of , and is uniquely determined up to a linear isomorphism.

Pontryagin class

The th Pontryagin class of a vector bundle is times the th Chern class of the complexification of the vector bundle. It is also in the th cohomology group of the base space involved.

Fundamental group

The fundamental group of an arcwise-connected set is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points at a given basepoint , under the equivalence relation of homotopy. The identity element of this group is the set of all paths homotopic to the degenerate path consisting of the point . The fundamental groups of homeomorphic spaces are isomorphic. In fact, the fundamental group only depends on the homotopy type of . The fundamental group of a topological space was introduced by Poincaré (Munkres 1993, p. 1).The following is a table of the fundamental group for some common spaces , where denotes the fundamental group, is the first integral homology group, denotes the group direct product, denotes the free product, denotes the ring of integers, and is the cyclic group of order .space ()symbolcirclecomplex projective space00figure eightKlein bottle-torusreal projective..

Vertex set

The vertex set of a graph is simply a set of all vertices of the graph. The cardinality of the vertex set for a given graph is known as the vertex count of .The vertex set for a particular graph embedding of a graph is given in the Wolfram Language using PropertyValue[g, VertexCoordinates] or GraphEmbedding[g]. Vertex sets for many named graphs are available via GraphData[graph, "VertexCoordinates"] (for the primary embedding) and GraphData[graph, "Embeddings"] (for all available embeddings).The vertex set of an abstract simplicial complex is the union of one-point elements of (Munkres 1993, p. 15).

Vertex scheme

If is a simplicial complex, let be the vertex set of . Furthermore, let be the collection of all subsets of such that the vertices , ..., span a simplex of . Then the collection is called the vertex scheme of (Munkres 1993, p. 15).

Van kampen's theorem

In the usual diagram of inclusion homomorphisms, if the upper two maps are injective, then so are the other two.More formally, consider a space which is expressible as the union of pathwise-connected open sets , each containing the basepoint such that each intersection is pathwise-connected. Then, the homomorphism induced by the inclusion map from the free product of the fundamental groups of the s to the fundamental group of , i.e.,(1)is surjective (Hatcher 2001, p. 43). In addition, if each intersection is pathwise-connected, then the kernel of is the normal subgroup generated by all elements of the form(2)where is the homomorphism induced by the inclusion , and so induces an isomorphism(3)

Universal cover

The universal cover of a connected topological space is a simply connected space with a map that is a covering map. If is simply connected, i.e., has a trivial fundamental group, then it is its own universal cover. For instance, the sphere is its own universal cover. The universal cover is always unique and, under very mild assumptions, always exists. In fact, the universal cover of a topological space exists iff the space is connected, locally pathwise-connected, and semilocally simply connected.Any property of can be lifted to its universal cover, as long as it is defined locally. Sometimes, the universal covers with special structures can be classified. For example, a Riemannian metric on defines a metric on its universal cover. If the metric is flat, then its universal cover is Euclidean space. Another example is the complex structure of a Riemann surface , which also lifts to its universal cover. By the uniformization theorem, the only possible..

Four lemma

A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are surjective, then is surjective. This lemma is closely related to the five lemma, whichis based on a similar diagram obtained by adding a single column.

Milnor's theorem

If a compact manifold has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if has negative curvature, then its fundamental group has exponential growth in the sense that grows exponentially, where is (essentially) the number of different "words" of length which can be made in the fundamental group.

Five lemma

A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are surjective, then is surjective. If and are bijective, the hypotheses of (1) and (2) are satisfied simultaneously, and the conclusion is that is bijective. This statement is known as the Steenrod five lemma.If , , , and are the zero group, then and are zero maps, and thus are trivially injective and surjective. In this particular case the diagram reduces to that shown above. It follows from (1), respectively (2), that is injective (or surjective) if and are. This weaker statement is sometimes referred to as the "short five lemma."

Tubular neighborhood

A tubular neighborhood of a submanifold is an embedding of the normal bundle () of into , i.e., , where the image of the zero section of the normal bundle is equal to .

Topologically conjugate

Two maps are said to be topologically conjugate if there exists a homeomorphism such that , i.e., maps -orbits onto -orbits. Two maps which are topologically conjugate cannot be distinguished topologically.

Extension problem

Given a subspace of a space and a map from to a space , is it possible to extend that map to a map from to ?

Spectral theorem

Let be a Hilbert space, the set of bounded linear operators from to itself, an operator on , and the operator spectrum of . Then if and is normal, there exists a unique resolution of the identity on the Borel subsets of which satisfiesFurthermore, every projection commutes with every that commutes with .

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