What is the homological connectivity of the complete abstract simplicial complex?

Question

I am trying to understand the concept of homological connectivity of an abstract simplicial complex. Specifically, I am trying to compute the homological connectivity of the complete abstract simplicial complex: the set $2^{X}$ where $X$ is some finite set. From this video I learned that, if we take any $d$-dimensional simplex containing its interior, then its $0$-th homological group is $\mathbb{Z}$, and all its other homological groups are trivial. If I understand correctly the definition of homological connectivity, this means that the homological connectivity of $2^X$ is infinity. Is this correct? Is there maybe a more intuitive way to see this (without going through all the computations)?

Answer 1

The thing you are calling the complete abstract simplicial complex is geometrically realized as nothing more than a ball of some dimension (whatever the size of the unique largest simplex is). This ball is contractible, and so has the homotopy type of a point, and so has infinite homological (indeed, even homotopical) connectivity.