Quantitative PL bordism

The proof of the bound in [CDMW18] built on the original work of René Thom [Tho53, Tho54] relating cobordism to the homotopy groups of the Thom space of the universal bundle over a Grassmannian. This Thom space is a specific finite complex $X$, and to prove the quantitative bound one must relate the Lipschitz constant of a nullhomotopic map $f:S^{N}\to X$ to the Lipschitz constant of its most efficient nullhomotopy. Even talking about Lipschitz constants requires a metric on $X$, but for an asymptotic computation the specific metric doesn’t matter because all the reasonable (Riemannian or piecewise Riemannian) metrics are Lipschitz homotopy equivalent.

The method of proof is quite general, and although this is not pointed out in that paper, it applies to oriented and non-oriented cobordism, as well as spin and complex cobordism and any similar variant. The reason is that in all such cases Thom’s method gives the appropriate reduction to the homotopy theory of finite polyhedra, and the spaces that arise are particularly simple from the point of view of rational homotopy theory.

However, this method does not apply to PL cobordism, because the classifying spaces are not close relatives of compact Lie groups. While they are abstractly homotopy equivalent to spaces with finite skeleta, we do not know of a natural way of representing them as such. This paper studies the complexity of PL nullcobordisms, with an eye towards developing techniques that can apply in situations where the traditional methods of geometric topology (such as classifying spaces, h-principles, etc.) reduce geometric problems to the homotopy theory of infinite complexes.

For infinite complexes, one can choose metrics that are inequivalent at large scales, and different choices can lead to very different measures of complexity. For example, a finite-type model for $K(\mathbb{Z},n)$ and a simplicial group model with infinitely many simplices in each dimension lead to very different “quantitative cohomology theories”. A related situation arises in K-theory, whose concrete manifestation is the following. Suppose one is interested in vector bundles of dimension $n$ over a finite complex $X$ of lower dimension. The number of such bundles which admit a connection with bounded curvature $K$ grows as a polynomial in $K$ whose degree is related to the cohomology of $X$. However, the naturally isomorphic set of stable vector bundles over $X$ can be realized by bundles (of varying dimension) with a uniform bound on curvature; if one takes curvature as the measure of complexity, then infinitely many classes are realized with bounded complexity.

Now, let us be a bit more precise. A reasonable notion of the complexity of a PL manifold $M$ is the number of simplices in a triangulation. We could consider the problem: given a PL $k$-manifold $M$ with $V$ simplices, how many simplices are contained in the simplest $W$ whose boundary is $M$?

It is true, but not completely obvious, that the complexity of a nullcobordism can be bounded by a recursive (i.e. computable) function of the complexity of $M$. This is equivalent to the statement that the PL bordism problem is algorithmically decidable, and that follows from the deep work that is explained e.g. in [MM79, Ch. 14]. One basic step in the analysis is the structure of G/PL, which is proved using surgery theory and deduced from the Poincaré conjecture. The algorithmic nature of these ingredients seems quite difficult to unravel.

One would have been led directly to the Poincaré conjecture in thinking about this question when considering whether a candidate $M$ that someone hands you is actually a PL manifold. Maybe your interlocutor is playing a joke on you? You can be sure that $M$ is a homology manifold, but it is much harder to know that $M$ is a manifold—you would need to see that the links of simplices are spheres. Checking that a manifold is actually a sphere, if you think it should be, is exactly what the Poincaré conjecture is about.

Of course, one might well produce a polyhedron bounded by $M$ and give it to the interlocutor, who can then worry about whether what you produced is a manifold. (Fair is fair.) So, despite first impressions, it is not quite necessary to complete an analysis of the algorithmic complexity of the Poincaré conjecture to handle the bordism question.

Regardless, we do not know how to deal with the PL bordism problem for arbitrary triangulations (although we do have a program that, if it succeeds, would give a tower of exponentials bound for the rise in complexity). At the same time, there is no indication that the true answer is any more nonlinear than it is for smooth bordism.

Instead we consider the case of PL manifolds of bounded geometry. That is, we consider PL manifolds with some bound on the number of simplices incident to each vertex, and build cobordisms constrained by a similar, perhaps larger bound. This condition is similar to the smooth locally bounded geometry condition in that the amount of “local information” is bounded. In fact, there is a closer correspondence: any smooth manifold of bounded geometry has a bounded geometry triangulation at scale $\sim 1$, and any smoothable PL manifold of bounded geometry admits a smoothing with locally bounded geometry and with simplices of bounded volume; see Appendix A.

Moreover, in every dimension $d$, there is an $L(d)$ such that every PL $d$-manifold is PL homeomorphic to one with bounded geometry of type $L(d)$. (However, given a manifold with $V$ simplices, the number of subdivisions required to implement a PL isomorphism with such a manifold grows faster than any computable function of $V$.) So bordism of PL manifolds with bounded geometry is equivalent, as far as pure topology is concerned, to bordism of PL manifolds.

As in the smooth case, our methods work in a number of settings: they extend from PL manifolds to pseudomanifolds whose singularities come from a prescribed family (see §4.1 for a precise definition), as well as to bordism homology of spaces other than a point. The full statement of our main theorem is therefore more general than Theorem 1.1:

Note that the constants depend in an inexplicit way on the bound on geometry. This is a point of contrast with the smooth category, where we can always change the bound on geometry by rescaling, which automatically scales the nullcobordism as well.

As in [CDMW18], we follow the usual proof of the classification of manifolds up to cobordism, bounding the complexity of each step. The main additional technical challenge stems from the fact that while classifying spaces for PL cobordism theories exist by Brown representability, unlike in the smooth case, there are no nice explicit models of finite type that we can use to construct our classifying maps. We resolve this by building the models we need, but these models may not actually be homotopy equivalent to the relevant classifying spaces; they do, however, have enough topological resemblance for our purposes.

Let $V$ be the number of $k$-simplices of the manifold $X$. We proceed as follows:

1. (1)

   Embed $X$ in $S^{n}$, with some control over the shape of a regular neighborhood. Here we can choose any sufficiently large $n$.

2. (2)

   This induces an $\varphi(V)$-Lipschitz map from $S^{n}$ to a fixed compact subspace of a kind of Thom space for $\mathcal{M}$-manifolds.

3. (3)

   One constructs a $\psi(\varphi(V))$-Lipschitz extension of this map to $D^{n+1}$, with image in a larger fixed compact subspace of this Thom space.

4. (4)

   From a simplicial approximation of this nullhomotopy, one can extract an $\mathcal{M}$-manifold embedded in $D^{n+1}$ which fills $X$ and whose number of simplices is bounded by the number of simplices in the approximation and is therefore $O(\psi(\varphi(V))^{n+1})$.

5. (5)

   After the previous step, the bound on the local geometry of the resulting filling $W$ depends on $n$. We tidy up by applying local modifications to $W$, making the bound on the geometry independent of $n$ at the expense of increasing the number of simplices by an additional multiplicative constant.

The final estimate depends on bounds on the functions $\varphi$ and $\psi$. Our results from the appendix to [CDMW18] suffice to show that

and we will show that $\psi(L)$ is at most linear. This gives an overall bound of $O(V^{1+O(1/n)})$ on the size of the resulting filling; since $n$ can be arbitrarily large, this gives our desired bound.

The most obvious such category is usual PL manifolds, with no singularities. Here we remark upon a prior result of F. Costantino and D. Thurston:

Note that in this theorem, $M$ is not required to have bounded geometry. However, in dimension 3 this doesn’t matter, and our main theorem implies a stronger result:

Notice that this strategy generalizes to higher dimensions: if every bounded geometry triangulated $S^{k}$ of volume $V$ can be filled with a bounded geometry disk of volume $F_{k}(V)$, then every $n$-manifold $M$ with a triangulation of unbounded geometry and volume $V$ has a triangulation of bounded geometry and volume

However, we currently have no estimates on the functions $F_{k}$ for $k\geq 3$. Finding such estimates seems to be a worthwhile and difficult problem. In a future paper we will show that $F_{7}$ is at least quadratic, the first nonlinear lower bound for this type of problem.

The next most obvious application of our theorem after PL manifolds is to pseudomanifolds with arbitrary singularities. Notice that the corresponding problem with unbounded geometry is trivial, as for any pseudomanifold one can fill it by taking the cone. On the other hand, restating our main theorem in this case gives the following new result:

This fact may be useful in the study of high-dimensional expanders, random complexes, and related topics.

There are a number of other categories of manifolds with singularities whose bordism theory is relevant in geometric topology. These include Witt spaces [Sie83, Gor84], whose bordism homology theory is closely related to KO-theory. Many other examples are discussed in [Fri15, §5.2.1].

Some such examples, including Witt spaces, have infinitely generated bordism groups. In this case, the corresponding classifying spaces do not have finite type. However, for a given bound on geometry, bordism classes are classified by a finite subspace. In other words, one can find bordism classes which require arbitrarily high local complexity, as expressed by the topology of links of simplices. Contrast this with the case of PL manifolds, in which all links have the same topology. Our main theorem still holds as written in such a setting, although for any given bound $L$ on the geometry, we are really only considering part of the bordism theory.

This suggests that, unlike in the case of PL manifolds, the problem of finding null-bordisms for Witt spaces without bounded geometry is fundamentally different.

On the other hand, note that if, for example, one is interested in Witt null-bordisms of non-singular PL manifolds, this can be done with finitely many topological link types, even without a bounded geometry assumption on the original manifold.

One area of application of these ideas is, in principle, to the connection between “secondary” invariants and complexity. The most famous secondary invariants—the $\eta$-invariants of Atiyah–Patodi–Singer [APS75, APS73, Wal99]—and their $L^{2}$ analogues due to Cheeger and Gromov are linearly bounded as a function of the volume of a manifold with bounded geometry [CG85]. These invariants are defined using the signature of a nullcobordism, and indeed, this inequality was Gromov’s original motivation for suggesting that cobordisms may have linear volume. Since our volume estimates are superlinear, they do not recover the optimal asymptotics of Cheeger and Gromov.

Cha [Cha16], in dimension 3, and Lim and Weinberger [LW23], in all dimensions, have proven an inequality for the Cheeger–Gromov $\rho$-invariant in terms of the number of simplices by a mixture of algebraic and geometric methods. The followup paper of Cha and Lim [CL] makes essential use of a non–locally finite classifying space for proving such estimates, and strongly suggests the possibility of extending the results of this paper beyond bounded geometry.

For other invariants, our results do provide a new estimate. For example, the bordism homology group $\Omega_{k}^{\mathcal{M}}(B\Gamma)$, when $\mathcal{M}$ is the class of Witt spaces, is used to define higher $\rho$-invariants of manifolds with fundamental group $\Gamma$ [Wei99]. If $B\Gamma$ is of finite type, our results can then be used to bound these higher $\rho$-invariants, and therefore (in certain situations) to bound the number of manifolds of bounded geometry homotopy equivalent to a given one. On the other hand, if $B\Gamma$ is not of finite type, we get an important example of a situation in which our results do not apply.

We first specify the necessary conditions on the category $\mathcal{M}$ that are required for bordism to make sense. These axioms are similar to those given in [BRS76, §IV.3]. A category of manifolds-with-singularity is specified by a family $\{\mathcal{L}_{n}\}_{n\in\mathbb{N}}$ of sets of $(n-1)$-dimensional simplicial complexes which constitute permissible links of vertices in a closed $n$-dimensional $\mathcal{M}$-manifold. These sets must satisfy:

1. (1)

   $\mathcal{L}_{n}$ is closed under PL isomorphism.

2. (2)

   Each member of $\mathcal{L}_{n}$ is a closed $\mathcal{M}$-manifold (that is, its links are contained in $\mathcal{L}_{n-1}$).

3. (3)

   If $P\in\mathcal{L}_{n}$, then $SP\in\mathcal{L}_{n+1}$.

4. (4)

   If $P\in\mathcal{L}_{n}$, then $cP\notin\mathcal{L}_{n+1}$.

The last axiom means that there is a notion of $\mathcal{M}$-manifold with boundary and that the boundary of such an object is well-defined. Namely, an $n$-dimensional $\mathcal{M}$-manifold with boundary is a simplicial complex whose links lie in $\mathcal{L}_{n}$ or $c\mathcal{L}_{n-1}$, and its boundary is the subcomplex of points whose links lie in $c\mathcal{L}_{n-1}$. Axiom 3 ensures that if $M$ is an $\mathcal{M}$-manifold, then $M\times I$ is a $\mathcal{M}$-manifold with boundary.

An additional “regularity” axiom, which is not strictly necessary, guarantees that $\mathcal{M}$-manifolds are pseudomanifolds:

1. (5)

   $\mathcal{L}_{1}=\{S^{0}\}$.

The construction of the Thom space is straightforward: it is patched together out of pieces which encode possible local behaviors of a $k$-dimensional $\mathcal{M}$-manifold embedded simplexwise linearly in $\mathbb{R}^{n}$. To be precise, $\operatorname{Th}\mathcal{M}(n,k)$ is a simplicial complex glued out of subcomplexes $K_{P,f,N}$ corresponding to triples $(P,f,N)$, where:

• •

  $P$ is a permissible link in $\mathcal{M}$.

• •

  $f:cP\to\mathbb{R}^{n}$ is a linear embedding of the cone on $P$. Let $p_{0}$ be the cone point of $cP$.

• •

  $N$ is a neighborhood of $f(p_{0})$ which is equipped with a linear triangulation transverse to $f$ and such that $f^{-1}(N)\cap P=\emptyset$.

• •

  For every $k$-simplex $\Delta\subset P$, the subcomplex $N_{c\Delta}\subseteq N$ defined by

  $$N_{c\Delta}=\bigcup\{\text{simplices }\sigma\subset N\mid f^{-1}(\sigma)\subseteq\operatorname{int}\operatorname{st}(c\Delta)\}$$

  satisfies $N_{c\Delta}\cap f(c\Delta)\neq\emptyset$. In other words, there is at least one simplex of $N$ which intersects $f(c\Delta)$ nontrivially and does not intersect $f(c\Delta^{\prime})$ for any simplex $\Delta^{\prime}\subset P$ which does not contain $\Delta$. This condition can be thought of as ensuring that that $N$ has a nice simplicial projection to the cell complex dual to $f(cP)$.

We identify two triples $(P,f,N)$ and $(P,f^{\prime},N^{\prime})$ if $f$ and $N$ differ from $f^{\prime}$ and $N^{\prime}$ by the same translation.

Given such a triple, let $N_{\infty}\subset N$ be the subcomplex consisting of simplices disjoint from $f(P)$. Then $\operatorname{Th}\mathcal{M}(n,k)$ is glued out of complexes $K_{P,f,N}\cong N\cup cN_{\infty}$. Given two such subcomplexes $K_{(P,f,N)}$ and $K_{(P^{\prime},f^{\prime},N^{\prime})}$, if there are vertices $v\in P$ and $v^{\prime}\in P^{\prime}$ such that $f(cv)=f^{\prime}(cv^{\prime})$ and $N_{cv}=N_{cv^{\prime}}$, then the subcomplexes

are identified in $\operatorname{Th}\mathcal{M}(n,k)$. Moreover, all the cone points of all the $cN_{\infty}$ are identified (this is the “point at infinity” in the Thom space).

Optionally, we can make this into a Thom space for oriented $\mathcal{M}$-manifolds by adding orientation data for top-dimensional simplices of $cP$ to the triple $(P,f,N)$, and require that this data match in order to identify subcomplexes.

Notice that $\operatorname{Th}\mathcal{M}(n,k)$ occurs as a subcomplex in $\operatorname{Th}\mathcal{M}(n+1,k+1)$. This is because for any subcomplex $K_{(P,f,N)}$ of $\operatorname{Th}\mathcal{M}(n,k)$, $\operatorname{Th}\mathcal{M}(n+1,k+1)$ contains a subcomplex $K_{P^{\prime},f^{\prime},N^{\prime}}$ corresponding to the triple

Then $K_{(P,f,N)}$ includes into $K_{P^{\prime},f^{\prime},N^{\prime}}$ as the subcomplex $K_{v}$ for $v=(\text{cone point},0)\in\partial(cP\times[0,1])$. These inclusions respect the gluings that produce the complexes as a whole. If using oriented simplices, we require that $cP$ face in the positive direction in $cP\times[0,1]$.

Finally, we must show that these complexes are indeed Thom spaces in the sense we need:

Here we describe the modifications to the construction of $\operatorname{Th}\mathcal{M}(n,k)$ required to encode bordism classes of maps from $\mathcal{M}$-manifolds to a simplicial complex $Y$. We define the space $(\operatorname{Th}\wedge Y)\mathcal{M}(n,k)$ as follows. We use the same subcomplexes $K_{(P,f,N)}$, but we index them by an additional datum, a simplicial map $g:cP\to Y$. Given two vertices $v\in P$ and $v^{\prime}\in P^{\prime}$ such that $f(cv)=f^{\prime}(cv^{\prime})$ and $N_{v}=N_{v^{\prime}}$, we identify $K_{v}\subset K_{(P,f,N,g)}$ and $K_{v^{\prime}}\subset K_{(P^{\prime},f^{\prime},N^{\prime},g^{\prime})}$ if in addition $g|_{\operatorname{st}(cv)}=g^{\prime}_{\operatorname{st}(cv^{\prime})}$ under the obvious identification. Once again, all the cone points of all the $cN_{\infty}$ are identified.

Note that $(\operatorname{Th}\wedge Y)\mathcal{M}(n,k)$ comes with a natural projection map

which is finite-to-one if $Y$ is a finite complex.