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==Definition==

Below are given a number of equivalent definitions of the first homology group of a [[topological space]], using different homology theories. All these homology groups turn out to be isomorphic, via obvious choices of isomorphisms.

===Singular homology definition===

This definition is a particular case of the definition of [[singular homology]].

For a topological space <math>X</math>, the first homology group <math>H_1(X)</math> is defined as the quotient <math>Z_1(X)/B_1(X)</math> where the groups are defined as follows:

* We define a ''singular 1-simplex'' as a [[continuous map]] from the [[closed unit interval]] to <math>X</math>. In other words, a singular 1-simplex is a path.
* We define the singular 1-chain group <math>C_1(X)</math> as the free group with generators the singular 1-simplices. The elements of this singular 1-chain group, called singular 1-chains, and are defined as formal <math>\mathbb{Z}</math>-linear combinations of singular simplices.
* We define the ''boundary'' of a singular 1-chain <math>\sum a_if_i</math>, where <math>f_i</math> are simplices and <math>a_i</math> are integers, as a formal sum of points in <math>X</math> given by <math>\sum a_i[f_i(1) - f_i(0)]</math>.
* The singular 1-cycle group <math>Z_1(X)</math> as the subgroup of <math>C_1(X)</math> comprising those singular 1-chains whose boundary is zero. In other words, it is those singular 1-chains such that ''adding up'' all their initial points gives the same result as adding up all their terminal points.
* The singular 1-boundary group <math>B_1(X)</math> is the subgroup comprising those singular 1-chains that arise as the sum of the singular simplices that bound a function from the 2-simplex to <math>X</math>.

The homology group <math>H_1(X)</math> is defined as <math>Z_1(X)/B_1(X)</math>.

More intuitively, each element of the homology group, called a ''homology class'', represents a choice of singular cycle (i.e., a formal sum of singular 1-simplices) up to adding or subtracting singular boundaries, i.e., those cycles that arise as the boundary of a 2-simplex.

===Simplicial homology definition===

===Cellular homology definition and the CW complex case===

First homology group - Revision history

Vipul: Created page with '==Definition== Below are given a number of equivalent definitions of the first homology group of a topological space, using different homology theories. All these homology g...'