Homological connectivity

X is homologically-connected if its 0-th homology group equals Z, i.e. ${\displaystyle H_{0}(X)\cong \mathbb {Z} }$, or equivalently, its 0-th reduced homology group is trivial: ${\displaystyle {\tilde {H_{0}}}(X)\cong 0}$. When X is a graph and its set of connected components is C, ${\displaystyle H_{0}(X)\cong \mathbb {Z} ^{|C|}}$ and ${\displaystyle {\tilde {H_{0}}}(X)\cong \mathbb {Z} ^{|C|-1}}$ (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. ${\displaystyle H_{1}(X)\cong 0}$.[1] When X is a connected graph with vertex-set V and edge-set E, ${\displaystyle H_{1}(X)\cong \mathbb {Z} ^{|E|-|V|+1}}$. 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 conn_H(X), is the largest k for which X is homologically k-connected. If all reduced homology groups of X are trivial, then conn_H(X) is defined as infinity. On the other hand, if all reduced homology groups are non-trivial, then conn_H(X) is defined as -1.

Some authors define the homological connectivity shifted by 2, i.e., ${\displaystyle \eta _{H}(X):={\text{conn}}_{H}(X)+2}$.[2]

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.: ${\displaystyle H_{1}(X;\mathbb {F} _{2})\cong 0}$.