Homological connectivity explained

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)\congZ

, or equivalently, its 0-th reduced homology group is trivial:

\tilde{H_0}(X)\cong0

For example, when X is a graph and its set of connected components is C,

H_0(X)\congZ^|C|

and

\tilde{H_0}(X)\congZ^|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)\cong0

For example, when X is a connected graph with vertex-set V and edge-set E,

H_1(X)\congZ^|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,

η_H(X):=conn_H(X)+2

.^[1] 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 F₂-homologically 1-connected if its 1st homology group with coefficients from F₂ (the cyclic field of size 2) is trivial, i.e.:

H_1(X;F_2)\cong0

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

Relation with homotopical connectivity

Hurewicz theorem relates the homological connectivity

conn_H(X)

to the homotopical connectivity, denoted by

conn_\pi(X)

For any X that is simply-connected, that is,

conn_\pi(X)\geq1

, the connectivities are the same: $\text_H(X) = \text_(X)$ If X is not simply-connected (

conn_\pi(X)\leq0

), then inequality holds: $\text_H(X)\geq \text_(X)$ but it may be strict. See Homotopical connectivity.

Notes and References

Aharoni. Ron. Berger. Eli. Kotlar. Dani. Ziv. Ran. 2017-10-01. On a conjecture of Stein. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. en. 87. 2. 203–211. 10.1007/s12188-016-0160-3. 119139740. 1865-8784.
Meshulam. Roy. 2003-05-01. Domination numbers and homology. Journal of Combinatorial Theory, Series A. 102. 2. 321–330. 10.1016/s0097-3165(03)00045-1. 0097-3165. free.
Adamaszek. Michał. Barmak. Jonathan Ariel. 2011-11-06. On a lower bound for the connectivity of the independence complex of a graph. Discrete Mathematics. en. 311. 21. 2566–2569. 10.1016/j.disc.2011.06.010. 0012-365X. free.
Linial*. Nathan. Meshulam*. Roy. 2006-08-01. Homological Connectivity Of Random 2-Complexes. Combinatorica. en. 26. 4. 475–487. 10.1007/s00493-006-0027-9. 10826092. 1439-6912.
Meshulam. R.. Wallach. N.. 2009. Homological connectivity of random k-dimensional complexes. Random Structures & Algorithms. en. 34. 3. 408–417. 10.1002/rsa.20238. 1098-2418. math/0609773. 8065082.
Cooley. Oliver. Haxell. Penny. Penny Haxell. Kang. Mihyun. Mihyun Kang. Sprüssel. Philipp. 2016-04-04. Homological connectivity of random hypergraphs. math.CO. 1604.00842.
Bobrowski. Omer. 2019-06-12. Homological Connectivity in Random Čech Complexes. math.PR. 1906.04861.