First homology group

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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.

Simplicial homology definition

Cellular homology definition and the CW complex case