Homological connectivity
# Homological connectivity

In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups.

## Definitions

### Background

X is homologically-connected if its 0-th homology group equals Z, i.e. $H_{0}(X)\cong \mathbb {Z}$, or equivalently, its 0-th reduced homology group is trivial: ${\tilde {H_{0}}}(X)\cong 0$.

* For example, when X is a graph and its set of connected components is C, $H_{0}(X)\cong \mathbb {Z} ^{|C|}$ and ${\tilde {H_{0}}}(X)\cong \mathbb {Z} ^{|C|-1}$ (see graph homology). 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. $H_{1}(X)\cong 0$.

* For example, when X is a connected graph with vertex-set V and edge-set E, $H_{1}(X)\cong \mathbb {Z} ^{|E|-|V|+1}$. Therefore, homological 1-connectivity is equivalent to the graph being a tree. Informally, it corresponds to X having no "holes" with a 1-dimensional boundary, 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).

### Connectivity

The homological connectivity of X, denoted $conn_H(X),$ is the largest k ≥ 0 for which X is homologically k-connected. Examples:

* If all reduced homology groups of X are trivial, then $conn_H(X)$ = infinity. This holds, for example, for any ball.
* If the 0th group is trivial but the 1th group is not, then $conn_H(X)$ = 0. This holds, for example, for a connected graph with a cycle.
* If all reduced homology groups are non-trivial, then $conn_H(X)$ = -1. This holds for any disconnected space.
* The connectivity of the empty space is, by convention, $conn_H(X)$ = -2.

Some computations become simpler if the connectivity is defined with an offset of 2, that is, $\eta _{H}(X):={\text{conn}}_{H}(X)+2$. The eta of the empty space is 0, which is its smallest possible value. The eta of any disconnected space is 1.

## Dependence on the field of coefficients

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 $F_2-homologically$ 1-connected if its 1st homology group with coefficients from $F_2$ (the cyclic field of size 2) is trivial, i.e.: $H_{1}(X;\mathbb {F} _{2})\cong 0$.

## Homological connectivity in specific spaces

For homological connectivity of simplicial complexes, see simplicial homology. Homological connectivity was calculated for various spaces, including:

* The independence complex of a graph;
* A random 2-dimensional simplicial complex;
* A random k-dimensional simplicial complex;
* A random hypergraph;
* A random Čech complex.

## Relation with homotopical connectivity

Hurewicz theorem relates the homological connectivity ${\text{conn}}_{H}(X)$ to the homotopical connectivity, denoted by ${\text{conn}}_{\pi }(X)$.

For any X that is simply-connected, that is, ${\text{conn}}_{\pi }(X)\geq 1$, the connectivities are the same:

${\text{conn}}_{H}(X)={\text{conn}}_{\pi }(X)$

If X is not simply-connected (${\text{conn}}_{\pi }(X)\leq 0$), then inequality holds:

${\text{conn}}_{H}(X)\geq {\text{conn}}_{\pi }(X)$

but it may be strict. See Homotopical connectivity.

