For a cup of coffee more

There is nothing new in this post. It is the sort of thing that most people would probably call folklore or obvious, but then again, sometimes it is cool to write it down. And who knows, perhaps someone finds it useful.

There are, on occasion, obvious concepts which lack a name. Perhaps because they have not proven useful yet. But then again, they are so very natural, they deserve to be discussed somewhere.

If X and Y are two topological spaces, perhaps manifolds, perhaps something else, then one remembers from topology what it means for them to be homeomorphic. We also know what it means to be PL homeomorphic¹, that is when the homeomorphism isn’t merely continuous, but piecewise linear. Still, it pays sometime to go back in time and look at the history of the subject. Incidentally, let me mention this marvelous book on the Hauptvermutung and its history I found recently by the late Yuli Rudyak².

Let’s fix a field $K$. Anyway, in some contexts, naturally objects appear that are only homeomorphisms “over K“. Consider, for instance, the following setting: Assume X is a triangulated manifold, and e an edge in it. It is a well known and classical result of Newman³ that (see also Nevo⁴ for an explicit reference), if

Theorem X is a PL manifold, e an edge in its interior that is valid, then the contraction of e in X is PL homeomorphic to X.

Here, valid is simply a requirement to ensure the contraction is a simplicial complex (otherwise we would leave the world of simplicial complexes). For what is to come, a subcomplex Y is valid or strongly induced in a simplicial complex X if the star of every simplex⁵ of X not in Y, intersected with Y, results in a simplex of Y.

Now, if X is merely a manifold, the result of contracting a valid interior edge is not necessarily homeomorphic, and the example goes back to the marvelous time of PL topology that was the seventies: Cannon and Edwards⁶, for instance, figured out that a triangulated integral homology manifold is a manifold if and only if the link of each vertex is simply connected. This is different from a PL triangulation, where the links have to be PL homeomorphic to spheres and balls.

This allows one to construct all sorts of miracles: Consider for instance the fact that any contracible manifold has an arc spine⁷, that is, it can be triangulated in a way that every simplex of it intersects a valid edge e in its interior. That this is possible is, of course, thanks to the Cannon-Edwards Theorem: e is a PL singular set of the triangulation.

Contracting that edge, then, results in an object that is not a manifold (again by Cannon-Edwards) if the boundary of the original was not a homotopy sphere to begin with (as with most contractible manifolds): The link of the unique interior vertex is now a non-simply connected homology sphere. Hence, the object is not a manifold. In particular, it is not homeomorphic to the manifold we had before the contraction⁸.

However, it is still homotopy equivalent to the original space. And it is what we call K-homeomorphic:
Consider again X and Y. Consider the local homology sheaves:

$\mathcal{L}^X_q(U) := H_q(X, X \setminus U; K), \qquad \mathcal{L}^Y_q(V) := H_q(Y, Y \setminus V; K)$

We call a map f from X to Y a K-homeomorphism if it induces an isomorphism

$f^{-1}\mathcal{L}^Y_q \longrightarrow \mathcal{L}^X_q$

of sheaves for all q. Of course, this notion has its drawbacks. For instance, the property of being K-homeomorphic is not an equivalence relation. It is not even symmetric.

Further, the restriction of a K-homeomorphism to a subspace is not a K-homeomorphism.

We can fix the first issue by declaring two topological spaces K-equivalent if they are related by a sequence of K-homeomorphisms. We have the following version of Newman’s theorem:

Proposition Consider M a triangulated K-homology manifold, and let Z denote a strongly induced K-acyclic subcomplex of the interior of M. Then the simplicial map induced by the contraction is a K-homeomorphism.

Of course, this is related to the notion of refinements. A refinement induces a PL homeomorphism. We say a simplicial complex A is a K-subdivision of a complex B if there exists a cellular cosheaf $\sigma\rightarrow B_\sigma\subset A$ that associates to every $\sigma$ in B of dimension k, a k-dimensional K-homology ball $B_\sigma$ that is a subcomplex of A, such that whenever x, y are faces of B,

Here, a K-homology ball is a K-homology manifold with the K-homology of a ball and nontrivial boundary, which in turn has the K-homology of a sphere. We have the following reformulation (or rather, generalization with the same proof) of a result of Barnette⁹

Theorem Any K-sphere of dimension d is a K-subdivision of the boundary $\partial\varDelta_{d+1}$ of the d+1-dimensional simplex $\varDelta_{d+1}$.

To drive that point home, let us observe the following corollary

Corollary Any K-ball is a K-subdivision of the simplex of the same dimension. In particular, any K-subdivision of a simplicial complex induces a K-homeomorphism.

In particular, any two K-homology spheres are K-equivalent, making this resemble a Poincaré conjecture/theorem of sorts.

Still, looking at topology “over K” is not totally flexible. Consider for instance the complement of the trefoil knot in the three sphere; it has the cohomology ring of $S^1 \times D^2$. Thinking about why these two are not K-equivalent is a cute exercise, and teaches you a baby version of the idea of Whitehead torsion, which, when first heard, sounds a little surprising perhaps: We pass to a cover. In the case of Whitehead torsion, you pass to the universal cover.

Here, we pass to the infinite cyclic cover. Lifting this to the (K-)cobordism induced by the K-equivalence, we would expect that they, too, are K-equivalent. But the infinite cyclic cover of the trefoil knot complement has nontrivial homology… and the infinite cyclic cover of $S^1 \times D^2$ does not. Hence, they are not K-equivalent.

So whats the point of all of this? Well, it turns out that the flexibility of K-equivalence is one way to demonstrate why spheres, when it comes to face numbers and face rings in particular, don’t care much about being PL or not. This in particular explains (and can be used to prove) many results such as the higher Heawood inequalities, and Grünbaum conjecture (see the end of my previous post). The key construction is this, which I call “many waists”:

Imagine you are in a triangulated K-homology sphere S of dimension 2k, and let D denote a subcomplex of dimension at most k-1. Does there exist a waist, that is, a codimension one K-homology sphere S‘ in S that contains D? The answer is yes, if one allowes for a deformation of S to a K-equivalent sphere $\widetilde{S}$. And this works even if one prescribes the K-equivalence not to affect D.

Anyway, given the fact that K-homeomorphisms are so flexible, I wonder whether they have have other uses.

1. webhomes.maths.ed.ac.uk/~v1ranick/papers/pltop.pdf ↩︎
2. www.worldscientific.com/worldscibooks/10.1142/9887#t=aboutBook ↩︎
3. Newman, A theorem in combinatorial topology, J. London Math. Soc. 6 (1931), 186–192 ↩︎
4. Nevo, Higher minors and Van Kampen’s obstruction, Math. Scandi., 101 (2007), 161-176. ↩︎
5. The star of a simplex F is the minimal subcomplex containing all simplices containing F. ↩︎
6. J. W. Cannon, The recognition problem: What is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), no. 5, 832–866. ↩︎
7. F. D. Ancel and C. R. Guilbault, Compact contractible n-manifolds have arc spines (n ≥
   5), Pacific J. Math. 168 (1995), 1–10. ↩︎
8. This seems to be “forgotten knowledge”, as it was formulated as a question for instance here: graphics.stanford.edu/courses/cs448b-00-winter/papers/dey_edge_contract.pdf ↩︎
9. David Barnette, Graph theorems for manifolds, Isr. J. Math. 16 (1973), 62–72 ↩︎