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 $0\le \mathrm{\ell }<k\le d$, $$\text{dim}({\tilde{H}}_k(X;\mathbb{R}))\le \sum _{\eta \in X(\mathrm{\ell })}\mid \{\lambda \in \text{Spec}({\tilde{L}}_{k-\mathrm{\ell }-1}(\text{lk}(X,\eta ))):\lambda \le \frac{(\mathrm{\ell }+1)(d-k)}{k+1}\}\mid \mathrm{.}$$ 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 $\mathrm{\ell }=k-1$. 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.