A combinatorial nerve theorem

Abstract

The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a space. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem. Our theorem enables us to compute the homology of a simplicial complex explicitly using the combinatorial information of its subcomplexes and their non-trivial intersections using discrete Morse theory. Suppose $X$ is a simplicial complex with subcomplexes $A_1, A_2, \dots ,A_k$ such that $X= \cup_{i=1}^{k}A_i$. Then the main theorem of this paper states that we can explicitly compute the homology of $X$ using the information of given gradient vector fields on $A_i$ for each $i \in [k]$, and on their possible non-trivial intersections. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realization, continuity or homotopy.