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 $$\sum a_if_i$$, 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.