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

)

≅

Z

{\displaystyle H_{0}(X)\cong \mathbb {Z} }

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

H

0

~

(

X

)

≅

0

{\displaystyle {\tilde {H_{0}}}(X)\cong 0}

.

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

H

0

(

X

)

≅

Z

|

C

|

{\displaystyle H_{0}(X)\cong \mathbb {Z} ^{|C|}}

and

H

0

~

(

X

)

≅

Z

|

C

|

−

1

{\displaystyle {\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

)

≅

0

{\displaystyle H_{1}(X)\cong 0}

.

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

H

1

(

X

)

≅

Z

|

E

|

−

|

V

|

+

1

{\displaystyle H_{1}(X)\cong \mathbb {Z} ^{|E|-|V|+1}}

. Therefore, homological 1-connectivity is equivalent to the graph being a tree.