TITLE: Can the rank of the homology group of an abstract simplicial complex be computed in polynomial time?
QUESTION [7 upvotes]: I want to write a function that does the following:
Input:

An integer $n$
A function $f$ that maps nonempty subsets of $\{1, \dots, n\}$ to "yes" or "no" such that (a) every singleton set gets "yes," and (b) if any set gets "yes" then all its subsets get "yes" too.
An integer $k \le n$

Jointly, these two inputs define an abstract simplicial complex.
Output:

The rank of the homology group $H_k$ of the simplicial complex.

I want a function that is polynomial in the value of $n$ (not the size of the bit representation of $n$) that solves this problem.  Can this be done?

REPLY [2 votes]: Homology can be computed in polynomial time in the number of simplices -- this is just a Smith normal form computation -- but that is not the same thing as polynomial in $n.$