Homological connectivity

In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. This property is related, but more general, than the properties of graph connectivity and topological connectivity. There are many definitions of homological connectivity of a topological space "X".
Definitions.
Basic definitions.
"X" is "homologically-connected" if its 0-th homology group equals Z, i.e. formula_1, or equivalently, its 0-th reduced homology group is trivial: formula_2. When "X" is a graph and its set of connected components is "C", formula_3 and formula_4 (see graph homology for details). Therefore, homological connectivity is equivalent to the graph having a single connected component, which is equivalent to graph connectivity. It is similar to the notion of a connected space.
"X" is "homologically 1-connected" if it is homologically-connected, and additionally, its 1-th homology group is trivial, i.e. formula_5. When "X" is a connected graph with vertex-set "V" and edge-set "E", formula_6. Therefore, homological 1-connectivity is equivalent to the graph being a tree. Informally, it corresponds to "X" having no 1-dimensional "holes", which is similar to the notion of a simply connected space.
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" for which "X" is homologically "k"-connected. If all reduced homology groups of "X" are trivial, then connH(X) is defined as infinity. On the other hand, if all reduced homology groups are non-trivial, then connH(X) is defined as -1.
Variants.
Some authors define the homological connectivity shifted by 2, i.e., formula_7.
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.: formula_8.
Homological connectivity in specific spaces.
Homological connectivity was calculated for various spaces, including:
See also.
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".