Commutativity

Homology Relation


Each diagram is a network or linear graph in which each vertex represents a group, and each oriented edge represents a homomorphism connecting the groups at its two ends.

A directed path in the network represents the homomorphism which is the composition of the homomorphisms assigned to its edges. Two paths connecting the same pair of vertices usually give the same homomorphism. This is called a commutativity relation. The combinatorially minded individual can regard it as a homology relation due to the presence of 2-dimensional cells adjoined to the graph.

If, at some vertex of the graph, two abutting edges are in line, one oriented toward the vertex and the other away, it is frequently the case that the image of the incoming homomorphism coincides with the kernel of the outgoing homomorphism. This property is called exactness. It asserts that the group at the vertex is determined, up to a group extension, by the two neighboring groups, the kernel of the incoming homomorphism, and the image of the outgoing homomorphism. Exact sequences of groups and homomorphisms occur throughout. Their algebraic properties are readily established, and are very convenient.

-S. Eilenberg and N. Steenrod

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