A note on PL handlebodies, the Hausmann trick and some homology spheres

There is a surprising lack of intuition for PL manifolds around, which always surprised me. And it turns out you can answer some questions. We stay in the category of PL manifolds throughout. The following question was asked by Gil Kalai to Ed Swartz some 15 years ago. Ed could not answer the question, and popularized here.

Question (Kalai) Are there different (homeomorphism types of) triangulated closed compact homology $(d-1)$ -spheres $H$ with $g_3=0$ (for any $d$ )?

Without going too deep into what $g_3=0$ means, we note quite simply that it is equivalent to saying that $H$ is boundary to a triangulated manifold $D$ that has no interior simplices of dimension $d-3$ .

The answer to this is yes. In fact, there are infinitely many (). For this purpose, we will make the following observation:

Theorem H. Let $d \ge 2$ . Let $M$ be a PL $d$ -manifold. The following are equivalent:

1. $M$ admits a PL handle decomposition into handles of index $\le k$ ,

2. $M$ admits a PL triangulation in which all $(d-k-1)$ -simplices are on $\partial M$ .

Both directions are quite easy, and probably folklore; we shall need the implication from 1. to 2. here, for the converse, consider the dual of the triangulation, and attach the cells one by one in order of increasing dimension. Ed conjectured the implication from 2. to 1., but interestingly seemed to believe that the answer to Gil’s question is no. Alas, this theorem implies that the answer to Gil’s question is yes.

The Hausmann trick So what we want is an infinite number of homology spheres that are

Boundary to homology disks $D$ constructed only from 0, 1, and 2 handles
Are of distinct homeomorphism type. We use their fundamental groups as invariants.

To this end, let us construct $D$ first. They are obtained as $d$ -dimensional handlebodies, $d\ge 6$ , constructed over twodimensional presentation complexes for distinct perfect groups with balanced presentations (meaning number of generators = number of relations).

This takes care of both requirements: First, note that we only use handles corresponding to cells of the presentation complex; these are of dimension two only.

Second, $\pi_1(\partial D)= \pi_1(D)$ by general position in dimension principle in dimension $6$ (the presentation complex, perturbed into general position, does not intersect itself and therefore can be homotoped to the boundary.)

So how do we get a perfect group with a balanced presentation? This is simple, and was taught to me by my friend Louis Funar: pick any finitely generated perfect group, and write its generators in terms of commutators, one for each: This obviously provides a perfect group again, a priori larger, but it has balanced presentation. Now, we can simply consider the direct product of such groups to get infinitely many. It remains to provide the

Proof of Theorem H We need a lemma

Lemma G. Let $k \ge 1$ be an integer. Let $C$ be a PL-embedded $(k-1)$ -complex in a PL $(d-1)$ -complex $N$ , where $k\le d$ . Let $T$ be a PL triangulation of $N$ such that $C$ is transversal to $T$ ( $j$ faces intersect $i$ faces in dimension $i+j-d+1$, or not at all). Then there is a subdivision $T'$ of $T$ such that

1. the neighborhood of $C$ inside $T'$ is regular, and $C$ is transversal to $T’$;

2. $T'$ is obtained from $T$ by stellarly subdividing only faces of dimension $\ge d-k$ .

Here, stellar subdivision is the subdivision obtained by removing a face, and coning over the boundary of the hole left; we can think of it as simply taking the cone over the neighborhood of the face in question, and forgetting about the part covered by it. In particular, if $N=\partial M$ , then this operation of stellarly subdividing at a face of dimension $s$ introduces an interior face of dimension $s$, but not of lower dimension.

Regular neighborhood means that the collection of faces incident to $C$ strongly PL deformation retracts onto $C$ . For this, it is enough to show that the new subdivision $T'$ has the following property:

If $v$ is any vertex of $T'$ , then the restriction of $C$ to the neighborhood (the star) $st_v N$ of that vertex of $T'$ is conical (it has a unique minimal face), and hence contractible.

For this, we can use the following observation: If $P$ is any polyhedron, and $A, B$ are disjoint polytopes inside it so that they intersect no common facet of $P$, then they can be seperated using stellar subdivisions at maximal faces. This is trivial, as a single such subdivision suffices up to PL homeomorphism.

We can now prove this statement by induction on the dimension of $N$ , assuming by induction that we have proven the statement for the codimension one skeleton of $N$ , and the restriction of $C$ to that skeleton.

Now, given $N$ , we therefore may assume that the statement holds for the boundary of $st_v N$ , and apply the observation. This finishes the proof of the lemma.

Now we can finish the proof of Theorem H, which we do by induction on dimension of the manifold. For this, we attach the handles one by one. In a single step, say from $M_\ell$ to $M_{\ell+1}$ , consider the attaching sphere $S$ for the handle PL embedded into $\partial M_\ell$ . By Lemma G we may assume that the neighborhood $U$ of $S$ is regular. Consider $K$
the cone over $U$ . This is not a manifold, but we can turn it into one by considering the tangent space at the conepoint. By induction on the dimension we can turn the link into a disk $K'$ without adding interior faces whose codimension exceeds the index of the handle. Set $M_{\ell+1}=M_{\ell} \cup K'$ That finishes the proof. See also here.