Unsolicited Nonsense

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. Also… let’s face it, a lot of PL/combinatorial topology is a mess, and things that used to be well known get lost.

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 contractible manifold of dimension at least 5 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, and has K-acyclic fibers. 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 (finite) zig-zag of K-homeomorphisms, that is, $X_0$ is K-equivalent to $X_n$ if we have

where each $\leftrightsquigarrow$ denotes a K-homeomorphism that can go in either direction. It is a cute but intruiging property. Let me state an apparently nontrivial problem:

Problem (strong factorization) Assume M and N are K-equivalent triangulated manifolds. Is there an X so that

Back to the plot. We have the following version of Newman’s theorem:

(Newman over K) 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 first homology… and the infinite cyclic cover of $S^1 \times D^2$ does not. Hence, they are not K-equivalent.

So what’s the point of all of this? Well, there is a classical nice theorem, let’s call it the regular neighborhood theorem: If M is a PL manifold of dimension 2k-1, and X a subcomplex of dimension k-1 (or less), then it is a classical fact of PL topology that X embeds into the boundary of its regular neighborhood. In other words, there is a PL homeomorphism M to M’ that is the identity on X such that the regular neighborhood is realized by the simplicial neighborhood N of X, and $\partial N$ contains a simplicial complex X‘ isomorphic and ambient isotopic to X.

Again, I say this is classic, and would probably point to the great olds like Whitehead (or specifically Hudson and Zeeman in this case¹⁰), but some parts of this have left common knowledge. For instance, for $\partial N$ to contain a copy of X as a simplicial complex, and not only some subdivision, I know no better reference than my own paper with Zuzka¹¹, though it is probably folklore.

In any case, it is a neat and useful construction. It is related to the following property of PL spheres, let us call it the theorem of waists: If S is a PL sphere of dimension 2k, and X a subcomplex of dimension k-1, then there exists a subdivision S’ of S that contains a waist for X, that is codimension one triangulated sphere H (a hypersurface sphere, or hypersphere if you like) that contains X as a subcomplex.

Now… I wanted something like this for K-homology manifolds (and in the case of the waist theorem, homology manifolds that are also homology spheres). This was all for the final parts of the g-conjecture paper here, to extend the methods from manifolds that are PL to arbitrary K-homology manifolds. But classical PL methods break down at PL singularities. I gave a way to do get around, but it was circumventing actually having a version of the regular neighborhood theorem for homology manifolds. So, I went back recently and thought about the combinatorial topology of translating the regular neighborhood theorem, and whether it could be done.

In other words, the fact that the manifold was PL singular, was not PL, was a bug that needed to be overcome.

Enter K-homeomorphisms, and thinking beyond manifolds and with K-homology manifolds and K-homeomorphisms, it becomes a feature. Because they allow you to extend the regular-neighborhood theorem and the theorem of waists, see also here. The waists theorem is Corollary 8.2 there, and I needed a more general version of the regular neighborhood theorem (Lemma 8.5/Proposition 8.12 in the manuscript). But let me take a moment to formulate and prove the regular neighborhood theorem for K-homology manifolds here, which is less complicated (and less general) than Lemma 8.5/Proposition 8.12.

Theorem Consider M a K-homology manifold of dimension 2k-1, and X a subcomplex of dimension k-1 in its interior. Then, there is a K-equivalent manifold M’ (and the equivalence can be chosen not to affect X) in which X is valid, such that the simplicial neighborhood N of X in M’ collapses to X¹² , and so that $\partial N$ contains an ambient isotopic copy of X.

Proof (sketch): Perform two biased derived subdivisions¹³ of M with respect to X. Now X is strongly induced. Let $\widetilde{N}$ denote the neighborhood of X, and observe that the fiber of the natural projection to X over a face F has vanishing K-homology below dimension k. Moreover, we can think of it as a diagram of spaces¹⁴ over the face poset of X itself, which is of height k-1 (and in particular no more than the homological connectivity of the fibers). Hence, we can contract acyclic subsets of these fibers to points (realizing this simplicially again by further biased derived subdivisions) and obtain a copy X’ of X in the boundary of the new neighborhood N. Because of the (Newman over K) Proposition, this induces a K-homeomorphism, as desired. $\square$

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

1. webhomes.maths.ed.ac.uk/~v1ranick/papers/pltop.pdf ↩︎
2. www.worldscientific.com/worldscibooks/10.1142/9887#t=aboutBook ↩︎
3. M. H. A. Newman, A theorem in combinatorial topology, J. London Math. Soc. 6 (1931), 186–192 ↩︎
4. E. 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 another case of “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. D. Barnette, Graph theorems for manifolds, Isr. J. Math. 16 (1973), 62–72 ↩︎
10. J. F. P. Hudson and E. C. Zeeman, On Regular Neighbourhoods, Proc. London Math. Soc. 14 (1964), 719–745 ↩︎
11. K. Adiprasito and Z. Patáková (2025), A higher dimensional version of Fáry’s theorem. Bull. London Math. Soc., 57: 1409-1414. ↩︎
12. This is a consequence of X being valid/strongly induced in S’. ↩︎
13. See my aforementioned paper with Zuzana, section 1.1. ↩︎
14. E.D. Farjoun (1987). Homotopy and homology of diagrams of spaces. In: Miller, H.R., Ravenel, D.C. (eds) Algebraic Topology. Lecture Notes in Mathematics, vol 1286. Springer, Berlin, Heidelberg. ↩︎