An eigenvalue interlacing approach to Garland's method

Let $X$ be a pure $d$-dimensional simplicial complex. For $0\le k\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let $\tilde{H}_k(X;\mathbb{R})$ be its $k$-dimensional reduced homology group with real coefficients. For $\sigma\in X$, let $\text{lk}(X,\sigma)$ be the link of $\sigma$ in $X$. For a matrix $M$, we denote by $\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that, for every $0\le \ell<k \le d$, $$\text{dim}(\tilde{H}_k(X;\mathbb{R}))\le \sum_{\eta\in X(\ell)}\left| \left\{ \lambda\in \text{Spec}(\tilde{L}_{k-\ell-1}(\text{lk}(X,\eta))) :\, \lambda\le \frac{(\ell+1)(d-k)}{k+1}\right\}\right|.$$ This extends the classical vanishing theorem of Garland, corresponding to the special case when the right hand side of the inequality is equal to zero, and a more recent result by Hino and Kanazawa, corresponding to the case $\ell=k-1$. A main new ingredient in our proof is an abstract version of Garland's local to global principle, which follows as a simple consequence of the eigenvalue interlacing theorem, and may be of independent interest.