Universal bounds for decompositions of PL manifolds

Abstract

The geometric dimension of a nonempty compact PL manifold $N$ is the smallest nonnegative integer $d$ for which $N$ admits a handle decomposition with all handles of index at most $d$. A PL filling of a compact PL manifold is a decomposition into compact codimension $0$ PL submanifolds, called sectors, with pairwise disjoint interiors and intersections described by standard local models. We advance Singhof's program of simplifying the sectors of a filling and their intersections. Specifically, for integers $m, \ell\ge 1$ and nonnegative integers $a_1\le\cdots\le a_\ell$, we prove that every closed PL $m$-manifold admits a filling into $\ell$ sectors whose nonempty $r$-fold intersections have geometric dimension at most $a_r$ for each $1\le r\le \ell$ if and only if \[ a_{λ_1}+\cdots+a_{λ_q}\ge m-\ell+1 \] for every partition $λ_1+\cdots+λ_q=\ell$ into positive integers. Under the same inequalities, any prescribed $\ell$-sector filling of the boundary of a compact PL $m$-manifold extends to a filling with the corresponding relative handle index bounds. Necessity follows from the cup-product obstructions of Courte, Moussard, Ren, and Zhou. To prove sufficiency, we develop compatible handle trades that simplify intersections in increasing order of depth. For nondecreasing tuples of nonnegative integers, our theorem resolves the PL classification problem posed by Courte, Moussard, Ren, and Zhou. In particular, we obtain the bound $(2,2,4)$ for $3$-sector fillings in dimension $6$ and improve the $5$-sector bound $(1,2,3,3,4)$ of Rubinstein and Tillmann to $(1,2,2,3,4)$ in dimension $8$.