Stacked Piecewise Manifolds Overview

• Stacked piecewise manifolds are triangulated PL homology manifolds constructed by iterative stacking operations that yield minimal face numbers and satisfy tightness criteria.
• Their construction leverages combinatorial stacking and handle additions to systematically control topology, neighborliness, and f‐vector minimization.
• Extremal invariants and algebraic formulations highlight their significance in testing lower bound theorems and classifying minimal triangulations in higher dimensions.

A stacked piecewise manifold is a triangulated piecewise-linear (PL) homology manifold, with or without boundary, constructed through a hierarchy of combinatorial stacking operations starting from the simplex or the boundary of a simplex. These objects play a fundamental role in the combinatorial theory of manifolds, providing extremal examples that saturate lower bound theorems for face numbers, and they exhibit deep connections to notions of tightness, neighborliness, and minimal triangulations. In higher dimensions, stackedness is closely tied to tightness: in dimensions $d\geq4$, a tight, homologically simple closed PL manifold is necessarily stacked. Generalizations, such as $k$-stacked (or $r$-stacked) manifolds, stratify this hierarchy, yielding a combinatorial stratification of the category of PL homology manifolds. The classification, enumerative invariants, and extremal properties of stacked piecewise manifolds have been clarified in a series of foundational works (Bagchi, 2014, Datta et al., 2014, Murai et al., 2012, Datta, 2015).

1. Definitions and Hierarchy of Stackedness

Let $\Delta$ be a finite simplicial complex of dimension $d$, viewed as a triangulation of a (homology) $d$-manifold (possibly with boundary) over a field $\Bbbk$.

• Stacked triangulated $(d+1)$-manifolds with boundary: $\Delta$ is stacked if every interior face of $\Delta$ has dimension at least $d$; equivalently, the $(d-1)$-skeleton of $\Delta$ coincides with that of its boundary: $$\mathrm{Skel}_{d-1}(\Delta) = \mathrm{Skel}_{d-1}(\partial\Delta)$$ (Datta et al., 2014, Datta, 2015).
• Stacked triangulated $d$-manifolds without boundary: $\Delta$ is stacked if it is the boundary of some stacked $(d+1)$-manifold with boundary. For spheres, stackedness coincides with being the boundary of a stacked ball (Bagchi, 2014).
• $k$-stacked (or $(r-1)$-stacked) triangulations: A $(d+1)$-manifold with boundary is $k$-stacked if its $(d-k)$-skeleton is the same as that of its boundary: $$\mathrm{Skel}_{d-k}(\Delta) = \mathrm{Skel}_{d-k}(\partial\Delta)$$. A closed $d$-manifold is $k$-stacked if it is the boundary of a $k$-stacked $(d+1)$-manifold (Murai et al., 2012, Bagchi, 2014).
• Locally stacked manifolds: For $d\geq4$, closed $d$-manifolds are stacked if and only if all vertex-links are stacked spheres (local stackedness) (Datta, 2015, Datta et al., 2014).

The hierarchy generalizes the notion of stacked polytopes and gives rise to a graded structure on triangulated manifolds.

2. Generating Operations and Classification

Stacked manifolds admit a recursive construction through emblematic combinatorial moves:

• Stacking operation (with boundary): Attach a $(d+1)$-simplex along a single $(d)$-face to a boundary face of a stacked ball increases the number of interior facets and vertices in a controlled fashion (Datta et al., 2014).
• Combinatorial handle addition (closed case): Remove two disjoint $(d-1)$-dimensional facets from a closed manifold and identify their vertices through an admissible bijection, preserving the manifold condition and stackedness (Datta et al., 2014, Datta, 2015).

Letting $\mathcal{H}_{d+1}(0)$ denote the boundary of the $(d+1)$-simplex, iterating handle additions yields a filtration $\mathcal{H}_{d+1}(k)$ of all stacked closed $d$-manifolds. Every stacked manifold (with or without boundary) is constructed through such stacking and handle moves (Datta et al., 2014).

Classification Theorem (Datta et al., 2014): For $d\ge2$, every connected stacked $d$-manifold (closed or with boundary) arises from a sequence of stacking (if with boundary) or handle additions (if closed) starting from the simplex or simplex boundary.

3. Enumerative and Topological Properties

Stacked manifolds occupy the minimal possible position in the face number lattice for manifolds of fixed dimension and topology, saturating generalized lower bound theorems:

• Face vector minimization: Stacked manifolds minimize non-topological entries in $f$-vectors, achieving the lower bound in Kalai's and Novik–Swartz's inequalities for the $g_2$-invariant (Datta, 2015).
• Extremal characterizations: For homology $d$-manifolds, stackedness is equivalent to the vanishing of enumerative invariants such as the $h^{\prime\prime}$-vector above a certain index; specifically, for $r$-stacked manifolds, $h^{\prime\prime}_r = 0$ (Murai et al., 2012).
• Missing faces: If $\Delta$ is $(r-1)$-stacked with boundary, then $\Delta$ has no missing $k$-faces for $k>r+1$, and $\beta_i(\Delta)=0$ for $i\ge r$ (Murai et al., 2012).
• Vertex-transitivity and minimality: Infinite families of stacked manifolds (e.g., the Datta–Singh series) exhibit vertex-transitive automorphism groups and are conjecturally minimal and strongly minimal triangulations (Datta, 2015).

4. Tightness, Neighborliness, and Stackedness

A triangulated $d$-manifold is $\Bbbk$-tight if for any induced subcomplex $Y\subseteq X$ and all $i$, the map $H_i(Y;\Bbbk)\to H_i(X;\Bbbk)$ is injective (Bagchi, 2014).

• Tightness criterion: Any $(k+1)$-neighborly, $k$-stacked $\Bbbk$-homology manifold with (or without, under orientability and $d\ne2k+1$) boundary is $\Bbbk$-tight (Bagchi, 2014).
• Characterization in high dimensions ($d\geq4$): Tightness plus vanishing intermediate homology ($\tilde H_i=0$ for $1$d$-manifold is locally stacked, and hence stacked (Datta et al., 2014, Datta, 2015).
• Extremal tight examples: Boundaries of $k$-stacked $(k+1)$-neighborly manifolds realize tightness and include constructions by Kühnel, Datta–Singh, and series arising from combinatorial handle theory (Bagchi, 2014, Datta, 2015).
• Combinatorial invariants and Morse-theoretic proof: Tightness is detected via comparison of Betti numbers and combinatorially defined vectors ($\mu$- and $\sigma$-vectors), with equality providing a Morse-theoretic characterization (Bagchi, 2014).

5. Algebraic and Enumerative Invariants

Stackedness and its refinements have algebraic formulations via the Stanley–Reisner face ring and associated invariants:

• $g$-vector and $\tilde{g}$-vector: For a stacked manifold, the $g$-vector and the algebraically defined $\tilde{g}$-vector (incorporating Betti numbers) coincide in the range $i<d/2$, and the entries are nonnegative (M-vectors) (Murai et al., 2012).
• Necessary conditions: For a connected orientable homology $(d-1)$-manifold with all vertex-links polytopal (having the weak Lefschetz property), $\Delta$ is $(r-1)$-stacked if and only if $\tilde g_r(\Delta)=0$ (Murai et al., 2012).
• Socle and algebraic lower bounds: The socle dimension of the quotient ring controls the sharpness of the lower bounds satisfied by stacked manifolds, with conjectures relating to strengthening existing inequalities (Murai et al., 2012).

6. Known Examples and Infinite Families

Stacked manifolds are realized by explicit combinatorial constructions, often yielding infinite families:

Dimension $d$	Vertices $f_0$	Manifolds / Series	Properties	Reference
$d$	$d+2$	Boundary of $(d+1)$-simplex	Standard stacked sphere, tight	(Datta, 2015)
$d$	$2d+3$	Kühnel's $K_{2d+3}$	Closed, neighborly, unique for $\tilde\beta_1=1$	(Datta, 2015)
$d$	$d^2+5d+5$	Datta–Singh series	Closed, neighborly, tight, cyclic automorphism	(Datta, 2015)
$2$	$n\equiv0,3,4,7\bmod{12}$	Neighborly closed surfaces	Classified, extremal, tight or $\mathbb{Z}/2$-tight	(Datta, 2015)
$3$	$9,29,49,89,109,\ldots$	Construction via graphs	Closed, neighborly, tight, strong minimality	(Datta, 2015)
$d\geq3$	$d^2+3d+1$	Datta–Singh boundary examples	Manifolds with boundary, $1$-stacked, $2$-neighborly	(Bagchi, 2014)

The class of stacked manifolds is closed under combinatorial handle addition and stacking operations, providing a method for generating all such manifolds.

7. Open Problems and Further Directions

• Classification in higher dimensions: Except for three infinite families and explicit surface/low-dimensional cases, there is limited knowledge of tight or stacked triangulated manifolds for $d\ge5$ (Datta, 2015).
• Relations between tightness and stackedness: It remains open whether every tight triangulation must be stacked (for $d\ge4$), or whether there exist non-combinatorial tight homology manifolds (Bagchi, 2014, Datta, 2015).
• Minimality conjectures: Every tight triangulated manifold is conjectured to minimize the number of vertices or even be strongly minimal (Datta, 2015).
• Lower bound conjectures: The generalized lower bound conjecture (GLBC) posits explicit inequalities involving $h$-vector components and Betti numbers, attaining equality exactly on locally stacked (stacked) triangulations (Murai et al., 2012).
• Extension to non-PL settings: Characterizing stackedness and its consequences in the smooth or general topological manifold categories remains an open subject (Datta et al., 2014, Datta, 2015).

Stacked piecewise manifolds are a central object in combinatorial and PL topology, providing a testbed for extremal, minimal, and algebraically rigid triangulations, with significant structural theorems and open classification problems (Bagchi, 2014, Murai et al., 2012, Datta et al., 2014, Datta, 2015).