Let \(G=(V,E)\) be a finite simple graph, where \(V\) and \(E\) denote the vertex set of \(V\) and the set of the edges of \(G\), respectively. A subset \(\sigma\subset V\) is called independent if there are no \(v,w\in \sigma\) such that \(\{v,w\}\in E\). The independence complex of a graph \(G=(V,E)\) is a simplicial complex whose vertex set is \(V\) and whose simplices are independent sets in \(V\). We denote by \(I(G)\) the independence complex of \(G\). For each pair \((n,k)\) of positive integers, let \(\Gamma_{n,k}=(V,E)\) denote the \((n\times k)\)-grid graph given by \begin{align*} V&=V(\Gamma_{n,k})=\{(x,y)\in \mathbb{Z}^2:1\leq k\leq n,1\leq y\leq k\}, \\ E&=E(\Gamma_{n,k})=\{\{(x,y),(z,w)\}:(x,y),(z,w)\in V, \vert x-z\vert+\vert y-w\vert =1\}. \end{align*}
In this paper, the authors investigate the homotopy type of the independence complex \(I(G)\) for \(G=\Gamma_{n,k}\). In particular, they prove that the independence complex \(I(\Gamma_{n,k})\) is homotopy equivalent to a wedge of spheres for \(k=4\) or \(k=5\) by using the fold lemma and its extended version.