# homology

Homology    is the general name for a number of functors  from topological spaces  to abelian groups  (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 homology  ) all coincide. There are other generalized homology , but I won’t consider those. There are also related cohomology  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.

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

 $\Delta_{n}=\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}|x_{i}\geq 0,\sum_{i=1}^{n}x% _{i}\leq 1\}$

of $\mathbb{R}^{n}$. The $0$-simplex is a point, the $1$-simplex a line segment  , the 2-simplex, a triangle  , and the 3-simplex, a tetrahedron   .

A singular $n$-simplex in a topological space $X$ is a continuous map $f:\Delta_{n}\to X$. 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 $C_{n}(X,\mathbb{Z})$.

Next, we define a boundary operator  $\partial_{n}:C_{n}(X,\mathbb{Z})\to C_{n-1}(X,\mathbb{Z})$. Intuitively, this is just taking all the faces of the simplex, and considering their images as simplices of one lower dimension  with the appropriate sign to keep orientations correct. Formally, we let $v_{0},v_{1},\ldots,v_{n}$ be the vertices of $\Delta_{n}$, pick an order on the vertices of the $n-1$ simplex, and let $[v_{0},\ldots,\hat{v}_{i},\ldots,v_{n}]$ be the face spanned by all vertices other than $v_{i}$, identified with the $n-1$-simplex by mapping the vertices $v_{0},\ldots,v_{n}$ except for $v_{i}$, in that order, to the vertices of the $(n-1)$-simplex in the order you have chosen. Then if $\varphi:\Delta_{n}\to X$ is an $n$-simplex, $\varphi([v_{0},\ldots,\hat{v}_{i},\ldots,v_{n}])$ is the map $\varphi$, restricted to the face $[v_{0},\ldots,\hat{v}_{i},\ldots,v_{n}]$, made into a singular $(n-1)$-simplex by the identification with the standard $(n-1)$-simplex I defined above. Then

 $\partial_{n}(\varphi)=\sum_{i=0}^{n}(-1)^{i}\varphi([v_{0},\ldots,\hat{v}_{i},% \ldots,v_{n}]).$

It is a simple exercise in reindexing to check that $\partial_{n}\circ\partial_{n+1}=0$.

For example, if $\varphi$ is a singular $1$-simplex (that is a path), then $\partial(\varphi)=\varphi(1)-\varphi(0)$. That is, it is the difference  of the endpoints (thought of as 0-simplices).

Now, we are finally in a position to define homology groups. Let $H_{n}(X,\mathbb{Z})$, the $n$ homology group of $X$ be the quotient

 $H_{n}(X,\mathbb{Z})=\frac{\mathrm{ker}\,\partial_{n}}{\mathrm{im}\partial_{n+1% }}.$

The association $X\mapsto H_{n}(X,\mathbb{Z})$ is a functor from topological spaces to abelian groups, and the maps $f_{*}:H_{n}(X,\mathbb{Z})\to H_{n}(Y,\mathbb{Z})$ induced by a map $f:X\to Y$ are simply those induced by composition of an singular $n$-simplex with the map $f$.

Some examples of homology groups:

 $H_{m}(\mathbb{R}^{n},\mathbb{Z})=\begin{cases}\mathbb{Z}&m=0\\ 0&m>0.\end{cases}$

This reflects the fact that $\mathbb{R}^{n}$ has “no holes”

Consider the space $\mathbb{R}P^{n}$, real projective space, which is $\mathbb{R}^{n+1}\setminus\{0\}$ modulo the relation    that $(x_{0},\ldots,x_{n})\equiv\lambda(x_{0},\ldots,x_{n})$ for every nonzero $\lambda$. For $n$ even,

 $H_{m}(\mathbb{R}P^{n},\mathbb{Z})=\begin{cases}\mathbb{Z}&m=0\\ \mathbb{Z}_{2}&m\equiv 1\pmod{2}\text{ or }n>m>0\\ 0&m\equiv 0\pmod{2},\,n>m>0\text{ or }m\geq n,\end{cases}$

and for $n$ odd,

 $H_{m}(\mathbb{R}P^{n},\mathbb{Z})=\begin{cases}\mathbb{Z}&m=0\text{ or }n\\ \mathbb{Z}_{2}&m\equiv 1\pmod{2}\text{ or }n>m>0\\ 0&m\equiv 0\pmod{2},\,n>m>0\text{ or }m>n.\end{cases}$
 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