HomologyMathworldPlanetmathPlanetmathPlanetmath is the general name for a number of functorsMathworldPlanetmath from topological spacesMathworldPlanetmath to abelian groupsMathworldPlanetmath (or more generally modules over a fixed ring). It turns out that in most reasonable cases a large number of these (singular homology, cellular homology, Morse homology, simplicial homologyMathworldPlanetmath) all coincide. There are other generalized homology , but I won’t consider those. There are also related cohomologyPlanetmathPlanetmath theories which serve the same purpose with slightly different machinery.

In an intuitive sense, homology measures “holes” in topological spaces. The idea is that we want to measure the topology of a space by looking at sets which have no boundary, but are not the boundary of something else. These are things that have wrapped around “holes” in our topological space, allowing us to detect those “holes.” Here I don’t mean boundary in the formal topological sense, but in an intuitive sense. Thus a loop has no boundary as I mean here, even though it does in the general topological definition. You will see the formal definition below.

Perhaps the simplest form of homology to visualize, and to work with in practice, is simplicial homology. It is based on computing the homology groups of a simplicial complex (generally a finite one). However, it is generally nontrivial to show that a space of interest is homeomorphicMathworldPlanetmath to a simplicial complex, and it can also be difficult to apply more advanced methods such as spectral sequencesMathworldPlanetmath when working with simplicial homology. Singular homology is similarMathworldPlanetmathPlanetmath: it is in some sense a continuousPlanetmathPlanetmath version of simplicial homology, and it does not suffer from these problems.

Singular homology is defined as follows: We define the standard n-simplex to be the subset


of n. The 0-simplex is a point, the 1-simplex a line segmentMathworldPlanetmath, the 2-simplex, a triangleMathworldPlanetmath, and the 3-simplex, a tetrahedronMathworldPlanetmathPlanetmath.

A singular n-simplex in a topological space X is a continuous map f:ΔnX. A singular n-chain is a formal linear combination (with integer coefficients) of a finite number of singular n-simplices. The n-chains in X form a group under formal addition, denoted Cn(X,).

Next, we define a boundary operatorMathworldPlanetmath n:Cn(X,)Cn-1(X,). Intuitively, this is just taking all the faces of the simplex, and considering their images as simplices of one lower dimensionMathworldPlanetmath with the appropriate sign to keep orientations correct. Formally, we let v0,v1,,vn be the vertices of Δn, pick an order on the vertices of the n-1 simplex, and let [v0,,v^i,,vn] be the face spanned by all vertices other than vi, identified with the n-1-simplex by mapping the vertices v0,,vn except for vi, in that order, to the vertices of the (n-1)-simplex in the order you have chosen. Then if φ:ΔnX is an n-simplex, φ([v0,,v^i,,vn]) is the map φ, restricted to the face [v0,,v^i,,vn], made into a singular (n-1)-simplex by the identification with the standard (n-1)-simplex I defined above. Then


It is a simple exercise in reindexing to check that nn+1=0.

For example, if φ is a singular 1-simplex (that is a path), then (φ)=φ(1)-φ(0). That is, it is the differencePlanetmathPlanetmath of the endpoints (thought of as 0-simplices).

Now, we are finally in a position to define homology groups. Let Hn(X,), the n homology group of X be the quotient


The association XHn(X,) is a functor from topological spaces to abelian groups, and the maps f*:Hn(X,)Hn(Y,) induced by a map f:XY are simply those induced by composition of an singular n-simplex with the map f.

From this definition, it is not at all clear that homology is at all computable. But, in fact, homology is often much more easily computed than homotopy groupsMathworldPlanetmath or most other topological invariantsPlanetmathPlanetmath. Important tools in the calculation of homology are long exact sequences, the Mayer-Vietoris sequence, cellular homology, spectral sequences, and homotopy invariance.

Some examples of homology groups:


This reflects the fact that n has “no holes”

Consider the space Pn, real projective space, which is n+1{0} modulo the relationMathworldPlanetmathPlanetmathPlanetmath that (x0,,xn)λ(x0,,xn) for every nonzero λ. For n even,

Hm(Pn,)={m=02m1(mod2) or n>m>00m0(mod2),n>m>0 or mn,

and for n odd,

Hm(Pn,)={m=0 or n2m1(mod2) or n>m>00m0(mod2),n>m>0 or m>n.
Title homology
Canonical name Homology
Date of creation 2013-03-22 13:14:41
Last modified on 2013-03-22 13:14:41
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 17
Author mathcam (2727)
Entry type Definition
Classification msc 55N10
Synonym singular homology
Related topic SimplicialComplex
Related topic GeometryOfTheSphere
Related topic BettiNumber
Related topic HomologyChainComplex
Related topic CohomologyGroupTheorem
Defines singular n-chain
Defines singular n-simplex