Fundamental Group, Singular Homology, Excision


Paris, Chennai

Algebraic Topology is about turning questions about the existence of topological objects to the corresponding algebraic objects. If an algebraic object does not exist, neither does the toplogical one, but since this process leads to information loss, existence of an algebraic object does not imply existence of the corresponding topological object.

Notation: we use to indicate homomorphism, to indicate isomorphism, and to indicate homotopy equivalence.

Homotopy

The maps are said to be homotopic, if there is a homotopy connecting them; this is denoted as . A homotopy that gives a deformation retraction of a space to space has the has the property 𝟙 for all .

The notation for two spaces and that are homotopy equivalent or have the same homotopy type is .

CW complexes

An orientable surface of genus can be constructed from a polygon with sides. The surface of the torus formed from this construction is termed a 2-cell, while the circles giving it structure are 1-cells.

Operations on spaces

Wedge sum of two spaces and , , is defined as quotient of the disjoint union obtained by identifying and as a single point. For instance, is homeomorphic to the figure '8'.

Smash product of two spaces and , , is defined as .

Paths and the fundamental group

Homotopy of paths leading from to can be written as , where and . Now,

Equivalence class of under equivalence relation of homotopy is called homotopy equivalence, and denoted by .

Composition or product path:

Paths that have the same starting and endpoint are called loops, and the point is called the basepoint. The set of all homotopy classes , with a common basepoint is called the fundamental group . This is the first in the sequence , where is replaced by n-dimensional unit cubes . is a group with respect to product .

A space is said to be simply-connected iff there is a unique homotopy class of paths connecting any two points in .

Induced homomorphisms

Induced homomorphisms between fundamental groups arise as a consequence of maps between spaces.

induces the homomorphism

taking loops based at to loops based at , such that .

Two basic properties of induced homomorphisms that make the fundamental group a functor:

  1. for the composition
  2. 𝟙𝟙

Fundamental group of a circle

The map sending sending integer to the homotopy class of loop based at is an isomorphism.

, , , and is called the lift of .

Brower's fixed-point theorem in two dimensions: Every continuous map has a fixed point with .

Borsuk-Ulam theorem in two dimensions: for every continuous map , there exists a pair of antipodal points in with .

if and are path-connected.

van Kampen theorem

The van Kampen theorem gives us a way of computing the fundamental group of spaces which can be decomposed into smaller spaces with known fundamental groups.

Here, N is the normal subgroup. This isomorphism holds if each intersection is path-connected.

Covering spaces

Covering spaces are a useful tool for computing fundamental groups. A covering space of is along with the map : there exists an open cover , such that, for every , is a disjoint union of open sets in , each of which is mapped homeomorphically by onto . need not be surjective.

In the helix example, we have , and the open cover can be taken to be disjoint arcs whose union is .

Universal cover: Smallest subgroup biggest covering space of .

Homology

is for and for . However, higher homotopy groups are difficult to compute, and we use homology groups instead. For the fundamental group of a circle, for .

Mayer-Vietoris sequences are the van-Kampen theorem equivalent in homology.

-Complexes

The union of all faces of simplices in is the boundary of , written as . The open simplex is the interior of .

A -complex on space is a collection of maps with each dependent on such that:

  1. The restriction is injective, and each point in the the image of exactly one such restriction.
  2. Each restriction of to a face of is one of the maps .
  3. is open iff is open in for each .

Simplicial homology

The task is to define simplicial homology groups of -complexes of . Let be the free abelian group with basis open n-simplicies of . The elements of are called n-chains, and can be written as , where .

To construct an -simplex, a cell is attached to the -simplex. The -simplex is given by , and the boundary is obtained by taking the faces of the -simplex: , where the over indicates that the vertex is deleted from the sequence. The boundary homomorphism is then given by:

Using , the composition

is zero, and this is more concisely written as . The defect in exactness of this sequence is expressed as the simplicial homology group . It follows that . Elements of are called cycles, and those of are called boundaries. Elements of are cosets of , and are called homology classes.

Singular homology

A singular n-simplex is simply a map ; it is termed "singular" because the simplex need not be a nice embedding and can have "singularities": all that's required is that be continuous.

Singular homology may be thought of a special case of simplicial homology: Let be the -complex with one n-simplex for each singular n-simplex with attached in the obvious way to -simplices of that are restrictions of to various -simplices of . Hence, the simplicial group is identical to the singular group .

For the chain sequence , the homology group is the quotient group , with singular n-cycles in the numerator and boundaries in the denominator. In homological algebra, this quotient is termed as defect, as an exact sequence would yield a vacuous quotient group.

A singular n-chain can always be written as with ; when we compute as a sum of simplices with signs , there may be some canceling pairs consisting of identical simplices with opposite sign.

Choose a maximal collection of canceling pairs to construct an n-dimensional simplex from a disjoint union of n-simplices , one for each , by identifying -dimensional faces corresponding to the cancelation pairs. Then, s induce a map . is a manifold locally homeomorphic to except at subcomplexes of dimension .

Corresponding to the decomposition of a space to its path-connected components , we have the isomorphism . Also, , for path-connected space . If is a point,

For any space , is the direct sum of s, one for each path component of . By definition since . Define a homomorphism by . Then, if is path-connected; hence induces the isomorphism .

It is convenient to define an augmented chain complex in which a point has a homology groups in all dimensions including zero, and this is termed as the reduced homology group :

Homotopy invariance

Two spaces that are homotopically equivalent have isomorphic homology groups. For map , the induced homomorphism is defined by composing each singular -simplex with to obtain .

The fact that the maps satisfy is expressed by saying that s define a chain map between the singular chain complex of to that of .

A chain map between chain complexes induces a homomorphism between the homology groups of the complexes: takes cycles to cycles and boundaries to boundaries, and hence induces .

If two maps are homotopic, they induce the same homomorphism .

Given a homotopy from to , we can define the prism operator as:

𝟙

is a chain homotopy between and since .

There are also induced homomorphisms for reduced homology groups, since .

Excision

Our goal is to find a relationship between spaces , subspace , and quotient space . Given a nonempty closed subspace which is a deformation retraction of the neighborhood of , the following long exact sequence is what we need:

where is the inclusion , and is the quotient map . Pairs of spaces satisfying this are called good pairs.

Analogous to what we had earlier:

  1. ; take so that : in the l.e.s of this pair are zero since is contractable. Exactness of this sequence then implies that maps are isomorphisms for , and that .
  2. Every map has a fixed point; Brower's fixed point theorem.

Relative homology groups

In relative homology, we ignore all singular chains of a subspace of a given space. Let be spaces, and be defined as the quotient group . Then, we have a chain of boundary homomorphisms:

The relation holds for this chain. The homology groups of this chain is termed as relative homology groups .

  1. Elements of are represented by relative cycles: n-chains such that .
  2. Relative cycle is trivial in iff there is a relative boundary for some .

For chain complexes , inclusion map , and quotient map , we have a chain complex of short exact sequences:

When we pass homology groups, this stretches out into the long exact sequence:

where denotes the homology group at , with defined similarly.

The description of the boundary map is: if class is represented by a relative cycle , then is the class of .

In the l.e.s of homology pairs , are isomorphisms for all .

Excision theorem

The excision theorem: For spaces such that closure of is contained within the interior of , the inclusion map induces the homomorphism for all .

Let be the set of subspaces of that form an open cover of , and be a subset of consisting of chains such that each has image in a subset of . The boundary map takes to so that forms a chain complex; this homology is written as .

The inclusion map is chain homotopic; that is, there exists such that are chain homotopic to identity. induces the isomorphism .

For good pairs , the map induces homomorphism .

To find explicit cycles representing generators of the infinite cyclic groups , and , replace with the equivalent ; viewing as a singular n-simplex, the identity map is a cycle generating . Let be the union of all but -dimensional faces of . Then,

To find a cycle representing , consider to be two -simplices and . Then, , viewed as a singular -chain, is the cycle representing since:

Since CW pairs are good pairs: let be subcomplexes of CW complex ; then, the inclusion induces the homomorphism for all .

Invariance of dimension: nonempty open sets are homeomorphic iff . For , we have

by excision. From the l.e.s of the pair ,

Since deformation retracts onto , we conclude that is for and zero otherwise. By the same reasoning, is for and zero otherwise. Since the homeomorphism induces the isomorphism

for all , we get .

Local homology groups can be used to tell when spaces are not locally homeomorphic at certain points. They are defined as, for space and , the homology groups . Indeed, for an open neighborhood of , excision gives the isomorphism , so these groups depend only on the local homology of , near .

Naturality

For an l.e.s to be natural means that, for the map , the following diagram commutes:

Equivalence of simplicial and singular homology

For a -complex, singular and simplicial homology groups are always isomorphic. Let be a -complex with the subcomplex ; is the -complex formed by the union of simplices of . Relative groups can be formed in the same way as for singular homology, via relative chains , and this yields a l.e.s of simplicial homology groups for the pair . There is a canonical homomorphism induced by , sending each -simplex of to its characteristic map .

The homomorphisms are isomorphisms for all and pair . The simplicial chain group is zero for , and is a free abelian group with basis the -simplicies of when . The corresponding singular homology groups can be computed by considering the map . Since the quotient group induces a homeomorphism of spaces , it induces isomorphisms on all singular homolgy groups. Thus, is zero for , and for , it is a free abelian group with basis relative cycles of the -simplicies of .

The five lemma: In the following diagram, if are isomorphisms, so is :

is finitely generated when is a -complex with finitely many n-simplicies since, in this case, the simplicial chain of the group is finitely generated, hence also is the subgroup of cycles and the quotient group . If we write as a direct sum of cyclic groups, then the number of summands are called the Betti numbers, and the integer specifying the order of finite cyclic summands are called torsion coefficients.

Degree

For the map , the induced homomorphism is a homomorphism of an infinite cyclic to itself, and must be of the form , where depends only on . is then called degree of , and denoted by . Some properties:

  1. 𝟙 since 𝟙𝟙.
  2. if is not surjective.
  3. If , then .
  4. since .
  5. if is a reflection of .
  6. The antipodal map 𝟙 has degree .
  7. If has no fixed points, then .

has a continuous field of nonzero tangent vectors iff is odd. Let be a mapping, assigning to each vector the vector , the tangent to at . Regarding as a vector at origin instead of , tangency just means that and are orthogonal at . Assume that at all ; then, replace by so that . Then, lie in the unit circle in the plane spanned by and . Letting go from to , we obtain the homotopy from the identity map of to the antipodal map 𝟙. Then, 𝟙𝟙, and hence must be odd.

is the only non-trivial group that can act freely on if is even. The action of a group on a space is defined as the group , the group of homeomorphisms , and the action is free if the homeomorphisms corresponding to each nontrivial element of has no fixed points. In the case of , the antipodal map generates a free action of . The degree of a homeomorphism must be , so the action of group on determines the degree function . This is a homomorphism since , and the action is free so it sends each element of to . Hence, for an even , has a trivial kernel.

Let , , be the function with preimage finitely many points . Then, let be the disjoint open neighborhoods of these points, and be the neighborhood of , the points mapped by . Then, , and we get the following commutative diagram:

so becomes multiplication by an integer called local degree at , written . .

To construct the map of any degree, let be the quotient map mapping each to the open ball in to a point, and identify all summands to a single sphere. Consider ; for almost all , we have identifying a single in each open ball . Since is a homeomorphism near , its local degree near is . Thus, we can produce with degree .

, where is the suspension of . Suspension preserves degree, and we get the following commutative diagram:

For , the suspension map maps only one point to the two poles of . This means that the local degree of at each of the poles must equal the global degree of : so, local degree of can be any integer when , just as the degree can be any integer when .