Morse shellings on products

Let $n$ be a non-negative integer. An $n$-simplex is the convex hull of $n+1$ points affinely independent in some real affine space. A face of a simplex is the convex hull of a subset of its vertices and we call it a facet when it has codimension one in the simplex. The standard $n$-simplex is the convex hull of the standard basis of $\mathbb{R}^{n+1}$. It will be denoted by $\Delta_{[n]}$, or sometimes just by $\Delta_{n}$, fixing an identification between its vertices and the set of integers $[n]=\{0,\dots,n\}$. Likewise, for every subset $J$ of $\{0,\dots,n\}$, we will denote by $\Delta_{J}$ the face of $\Delta_{[n]}$ whose vertices belong to $J$. A total order on the vertices of any simplex prescribes then an affine isomorphism with the standard simplex of the corresponding dimension.

A finite simplicial complex $K$ is a finite collection of simplices which contains all faces of its simplices and such that the intersection of any two simplices in this collection is a face of each of them, see [16, 5]. The dimension of such a complex is the maximal dimension of its simplices and it is said to be pure $n$-dimensional if all the simplices that are maximal with respect to the inclusion are of dimension $n$. Such a simplicial complex $K$ inherits a topology and the underlying topological space is usually denoted by $|K|$, see [16, 5]. When it gets homeomorphic to some manifold, any such homeomorphism is called a triangulation of the manifold.

The face vector or $f$-vector of an $n$-dimensional finite simplicial complex $K$ is the vector $f(K)=(f_{-1}(K),f_{0}(K),\dots,f_{n}(K))$, where for every $j\in\{0,\dots,n\}$, $f_{j}(K)$ denotes the number of $j$-simplices of $K$ while $f_{-1}(K)=1$ counts the empty set. Likewise, the $h$-vector $h(K)=(h_{0}(K),\dots,h_{n+1}(K))$ of $K$ is defined by the relation $\sum_{i=0}^{n+1}h_{i}(K)X^{n+1-i}=\sum_{i=0}^{n+1}f_{i-1}(K)(X-1)^{n+1-i}$, see [15, 21, 8, 24].

Let us finally recall that a finite simplicial complex is said to be shellable iff there exists an order $\sigma_{1},\dots,\sigma_{N}$ of its maximal simplices such that for every $i\in\{2,\dots,N\}$, $\sigma_{i}\cap\big{(}\cup_{j=1}^{N-1}\sigma_{j}\big{)}$ is non-empty of pure dimension $\dim\sigma_{i}-1$, see [12, 24] for instance. This means that the simplices $\sigma_{1},\dots,\sigma_{N}$ are not proper faces of any other simplex in $K$ and that any simplex in $\sigma_{i}\cap\big{(}\cup_{j=1}^{N-1}\sigma_{j}\big{)}$ is a face of a $(\dim\sigma_{i}-1)$-dimensional one in this intersection. It is convenient for us to allow this intersection for being empty, so that a shelling for us need not be connected, see Remark $2.16$ of [19] and §2.2.

We now recall the notions of tilings and shellings introduced in [17, 19].

Two basic tiles of same dimension and order are isomorphic to each other via some affine isomorphism. We denote by $T^{n}_{k}=\Delta_{[n]}\setminus\cup_{j=0}^{k-1}\Delta_{[n]\setminus\{j\}}$ the standard basic tile of dimension $n$ and order $k$, compare [17].

When $k\geq 1$, the dimension of $\mu$ ranges between $k-1$ and $n-2$ and the underlying basic tile has a unique face of dimension $k-1$, see [19]. Any basic tile is Morse and a Morse tile is said to be critical of index $k$ iff it is of order $k$ and its Morse face has minimal dimension $k-1$, while a closed simplex is critical of index zero. The other Morse tiles are said to be regular. We sometimes denote by $C^{n}_{k}$ (resp. $T^{n,l}_{k}$) a $n$-dimensional critical tile of index $k$ (resp. a $n$-dimensional Morse tile of order $k$ with $l$-dimensional Morse face), so that $C^{n}_{k}=T^{n,k-1}_{k}$. They are all isomorphic to each other via some affine isomorphism.

When the tiling uses only basic tiles, it is called an $h$-tiling, see [17, 19]. Of special interest is the case $S=\overline{S}$ where a finite simplicial complex is Morse tiled, but Definition 2.5 is more general and includes sets such as the triangulated handles $\stackrel{{\scriptstyle\circ}}{{\Delta}}_{k}\times\Delta_{n-k}$ of Theorem 1.4. The dimension of a tileable subset is the dimension of the underlying simplicial complex, that is the maximal dimension of the tiles in any Morse tiling. When all tiles have same dimension, the tiling is said to be pure dimensional. This is always the case on compact triangulated manifolds. Indeed,

In particular, $h_{0}(\tau)=c_{0}(\tau)$ and $h_{n+1}(\tau)=c_{n}(\tau)$. Recall that the $h$-vector $h(\tau_{K})$ of any $h$-tiling $\tau_{K}$ of a finite simplicial complex $K$ coincides with the $h$-vector of $K$ as soon as $h_{0}(\tau_{K})=1$ and in any cases, two $h$-tilings $\tau_{K}$ and $\tau^{\prime}_{K}$ of $K$ have same $h$-vector as soon as $h_{0}(\tau_{K})=h_{0}(\tau^{\prime}_{K})$, by Theorem $4.9$ and Corollary $4.10$ of [17].

The $c$-vector, or critical vector, of a Morse tiling encodes the Euler characteristic of the tiled simplicial complex. Indeed,

We finally recall the definition of Morse shellability given in [19].

A Morse tiled subset of $S$ is a union of tiles which is closed in $S$, see Definition $9$ of [19]. When the tiling uses only basic tiles, this notion of Morse shelling coincides with the classical notion of shelling, without the non-emptyness assumption though, see §2.1, Theorem $2.15$ and Remark $2.16$ of [19]. A finite simplicial complex, when equipped with a Morse tiling, carries discrete vector fields which are compatible with the tiling and in the case of a Morse shelling, any of these is the gradient vector field of a discrete Morse function in the sense of R. Forman [6], whose critical points are in one-to-one correspondence, preserving the index, with the critical tiles of the tiling, see Theorem $1.2$ of [19]. The Betti numbers of a Morse shelled finite simplicial complex thus get bounded from above by the number of critical tiles of the corresponding index of the shelling, see Corollary $1.5$ of [19].

In order to get a Morse shelling on the product of two Morse shelled complexes, we need the shellings to satisfy some tameness condition which we now introduce.

The first part of Example 2.12 shows that it is always possible to orient the one-skeleton of a finite simplicial complex $K$ in order to define in a compatible way a total order on the vertices of each of its simplices, turning it into an ordered simplicial complex in the sense of Definition $II.8.7$ of [5]. When $K$ is equipped with a Morse tiling, we would like in addition that for every Morse tile with underlying simplex $\sigma$ and non-empty Morse face $\mu$, the vertices of $\mu$ are the maximal ones among the ones of $\sigma$. Recall that the link $\textup{Lk}_{\sigma}(\mu)$ of $\mu$ in $\sigma$ is by definition the convex hull of the vertices of $\sigma\setminus\mu$. We thus would like that the edges between $\textup{Lk}_{\sigma}(\mu)$ and $\mu$ are oriented from $\textup{Lk}_{\sigma}(\mu)$ to $\mu$, see Figure 2.

Every $h$-tiling is tame by the first part of Example 2.12, since the second condition of Definition 2.13 is then empty and the first one satisfied. In fact, the order condition in Definition 2.13 provides a structure of ordered simplicial complex on $K$ given by Proposition 2.11, see Definition $II.8.7$ of [5], while the tameness condition requires some compatibility between this structure and the tiling.

The tameness property gets satisfied by any Morse tiling after one barycentric subdivision for example.

We may finally provide a criterium which ensures that a Morse tiling is tame.

For example, the condition in Proposition 2.15 gets satisfied if all tiles with non-empty Morse faces have disjoint underlying simplices.

Let $n$ be a non-negative integer, the involution $k\in\{0,\dots,n\}\mapsto n-k\in\{0,\dots,n\}$ induces the automorphism $v=(v_{0},\dots,v_{n})\in\mathbb{R}^{n}\mapsto\check{v}=(v_{n},\dots,v_{0})\in\mathbb{R}^{n}$.

For example, the real Betti numbers of closed connected oriented manifolds define palindromic vectors by Poincaré duality, see [16, 2]. The $h$-vectors of convex polytopes are palindromic as well, see [15, 22, 8, 24]. We are going to prove that $h$-vectors of Morse tilings are likewise often palindromic.

Example 3.4 exhibits in particular an $h$-tiling with non-palindromic $h$-vector on the triangulated two-sphere. Does there exist such $h$-tilings on the other even-dimensional spheres?

In the case of Morse tilings, we observe.

The tiling given by Example 3.6 contains two critical tiles of indices one and two which could be replaced by two regular tiles of order two to produce a tiling with palindromic $h$-vector. Such examples with non-palindromic $h$-vectors and $c$-vectors can be obtained in a similar way in higher dimensions. But we do not know any Morse tiled closed triangulated manifold with palindromic critical vector and non-palindromic $h$-vector.

Recall that the product of two simplicial complexes is not a simplicial complex, it is a polyhedral complex whose cells are products of two simplices. Such a product can nevertheless be triangulated in such a way that each product of two simplices becomes the union of simplices of the underlying affine space, see §$II.8$ of [5], [9, 20] and §4.3. We are going to consider such triangulations, which are primitive in the sense of Definition 3.7 and in fact associated to staircases, see §4.1 and [9].

Our main result is the following Theorem 3.8, where we denote by $uv$ the graded product of a vector $u=(u_{0},\dots,u_{n})$ of $\mathbb{R}^{n+1}$ with a vector $v=(v_{0},\dots,v_{m})$ of $\mathbb{R}^{m+1}$, that is the product of the corresponding polynomials, so that $uv=(w_{0},\dots,w_{n+m})$ where for every $k\in\{0,\dots,n+m\}$, $w_{k}=\sum_{j=0}^{k}u_{j}v_{k-j}$.

By triangulation of $S_{1}\times S_{2}$ we mean triangulations on the product of the underlying simplicial complexes. We do not guarantee Morse tileability for all primitive triangulations on this product, the ones for which we do by Theorem 3.8 are given by the tameness of the tilings, see §6.1. Let us also recall that all $h$-tilings are pure dimensional in the case of triangulated manifolds by Lemma 2.6. We deduce Theorem 1.2, namely.

When one of the tilings $\tau_{1}$, $\tau_{2}$ uses only regular tiles, the tilings given by Theorem 3.8 share the same property since $c(\tau_{1})c(\tau_{2})$ then vanishes. However, it is not supposed to be an $h$-tiling even if $\tau_{1}$ and $\tau_{2}$ are, for $c(\tau_{1})c(\tau_{2})$ has more than two non-vanishing entries in general. The following variant of Theorem 3.8 fills this gap.

Recall that the total orders on the vertices of the simplices of $K_{1}$ and $K_{2}$ are given by Proposition 2.11. If the $h$-tiling of $S_{1}$ (resp. $S_{2}$) contains an open simplex, then the $h$-tiling of $S_{2}$ (resp. of $S_{1}$) has in particular to be regular for Theorem 3.10 to apply. Now, if $S_{2}$ is the non-shellable tiling of $\partial\Delta_{2}$ given in the second part of Example 3.4, then Theorem 3.10 applies whatever $S_{1}$ is. We may compute the $h$-vector of $S_{1}\times S_{2}$ in this case.

In Theorem 3.11 again, $h(\tau)$ is then palindromic as soon as $h(S)$ is, or provided $S$ is homeomorphic to a closed manifold by Corollary 3.3, since $c(\tau)$ is palindromic. This result implies Theorem 1.1, showing that the product of a sphere and a torus of any dimensions carries $h$-tileable triangulations. More precisely, we deduce the following Corollary 3.13.

The $h$-vector $h^{n,m}$ of the $h$-tilings given by Corollary 3.13 has its own interest. By Theorem $4.9$ of [17], it does not depend on the choice of the tiling but differs from the $h$-vector of the underlying primitive triangulation, see Corollary $4.10$ of [17]. What is the asymptotic of $h^{n,m}$ as $m$ grows to $+\infty$? More precisely, what is the limit, as $m$ grows to $+\infty$, of the probability measure

where $\delta_{x}$ denotes the Dirac measure at $x\in\mathbb{R}$?

We also do not know which closed manifolds carry $h$-tileable triangulations. They have non-negative Euler characteristic by Lemma 2.9.

The sets $S_{1},S_{2}$ in Theorems 3.8 and 3.10 may just consist of single tiles. In fact, these results follow from this special case to which we devote this section.

These triangulations and shellings given by Theorem 3.14 are inherited from particular total orders on the vertices of the underlying simplices of $T_{1}$ and $T_{2}$, see §5.2.4. They all have same $h$-vector and satisfy the following duality property.

In Theorem 3.16, $\widecheck{h}(T_{1}\times T_{2})$ denotes the image of the $h$-vector $h(T_{1}\times T_{2})$ under the palindromic automorphism defined in §3.1.

Theorem 3.14 provides in particular Morse shellings on every handle, whatever its index is, where by handle of index $k$ and dimension $n$, we mean the product $\stackrel{{\scriptstyle\circ}}{{\Delta}}_{k}\times\Delta_{n-k}$ as defined in [19]. Such a Morse shelling has already been obtained in index $1$ and $n-1$, see Corollary $3.17$ of [19].

Recall that Theorem $1.4$ of [19], which provides Morse shelled triangulations on every closed three-manifold, has been obtained by successive attachments of such Morse shelled triangulated handles. We end this section by giving other remarkable shellings given by Theorem 3.14.

Recall that $T_{n}^{n}$ (resp. $T_{1}^{n}$) denotes a basic tile of dimension $n$ and order $n$ (resp. order one), see §2.2.

Let $m,n$ be two non-negative integers. We denote by $C(n,m)$ the set of increasing -not strictly increasing- functions $f:\{0,\dots,n\}\to\{0,\dots,m\}$ such that $f(n)=m$ and by $N(n,m)$ its cardinality. We recall that.

The space $C(n,m)$ is equipped with the involution $f\in C(n,m)\mapsto\check{f}\in C(n,m)$, where for every $j\in\{0,\dots,n-1\}$, $\check{f}(j)=m-f(n-1-j)$ and $\check{f}(n)=m$. Also, the lexicographic order on the n-tuples of integers induces a total order on $C(n,m)$, so that for every $f,g\in C(n,m)$, $f\leq g$ iff $(f(0),\dots,f(n-1))\leq(g(0),\dots,g(n-1))$. The minimum of $C(n,m)$ is thus a function which vanishes on $\{0,\dots,n-1\}$ while its maximum is the constant function equal to $m$.