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.^[1]

Definitions[]

Basic definitions[]

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.

Variants[]

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

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;^[3][4]
• A random 2-dimensional simplicial complex;^[1]
• A random k-dimensional simplicial complex;^[5]
• A random hypergraph;^[6]
• A random Čech complex.^[7]

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.

References[]