In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups.
[1] X is homologically-connected if its 0-th homology group equals Z, i.e.
, or equivalently, its 0-th reduced homology group is trivial:
X is homologically 1-connected if it is homologically-connected, and additionally, its 1-th homology group is trivial, i.e.
[1] In general, for any integer k, X is homologically k-connected if its reduced homology groups of order 0, 1, ..., k are all trivial.
Note that the reduced homology group equals the homology group for 1,..., k (only the 0-th reduced homology group is different).
The homological connectivity of X, denoted connH(X), is the largest k ≥ 0 for which X is homologically k-connected.
Examples: Some computations become simpler if the connectivity is defined with an offset of 2, that is,
[2] The eta of the empty space is 0, which is its smallest possible value.
The eta of any disconnected space is 1.
The basic definition considers homology groups with integer coefficients.
Considering homology groups with other coefficients leads to other definitions of connectivity.
For example, X is F2-homologically 1-connected if its 1st homology group with coefficients from F2 (the cyclic field of size 2) is trivial, i.e.:
For homological connectivity of simplicial complexes, see simplicial homology.
Homological connectivity was calculated for various spaces, including: Hurewicz theorem relates the homological connectivity
to the homotopical connectivity, denoted by
Meshulam's game is a game played on a graph G, that can be used to calculate a lower bound on the homological connectivity of the independence complex of G.