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Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
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Geometric interpretation of the lower central series for the fundamental group?
\ge G_i \ge\dots$$
In the case where $G = \pi_1(X)$ The first quotient $H^1 = G_0/G_1$ is well known to be the first homology group (which has well known geometric content). … As homology/cohomology of some other object on X? As something else? Group homology?
Question 4:
Is there a good source for these types of questions? …