First homology group

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 $X$, the first homology group $H_{1}(X)$ is defined as the quotient $Z_{1}(X)/B_{1}(X)$ where the groups are defined as follows:

• We define a singular 1-simplex as a continuous map from the closed unit interval to $X$. In other words, a singular 1-simplex is a path.
• We define the singular 1-chain group $C_{1}(X)$ 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 $\mathbb {Z}$-linear combinations of singular simplices.
• We define the boundary of a singular 1-chain $\sum a_{i}f_{i}$, where $f_{i}$ are simplices and $a_{i}$ are integers, as a formal sum of points in $X$ given by $\sum a_{i}[f_{i}(1)-f_{i}(0)]$.
• The singular 1-cycle group $Z_{1}(X)$ as the subgroup of $C_{1}(X)$ 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 $B_{1}(X)$ 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 $X$.

The homology group $H_{1}(X)$ is defined as $Z_{1}(X)/B_{1}(X)$.

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.