First homology group
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 , the first homology group is defined as the quotient where the groups are defined as follows:
- We define a singular 1-simplex as a continuous map from the closed unit interval to . In other words, a singular 1-simplex is a path.
- We define the singular 1-chain group 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 -linear combinations of singular simplices.
- We define the boundary of a singular 1-chain , where are simplices and are integers, as a formal sum of points in given by .
- The singular 1-cycle group as the subgroup of 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 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 .
The homology group is defined as .
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.