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Hello world.

I’m going to start sparsely writing down my thoughts which I think are interesting, but which are not quite put together enough to stick in a paper. I’ll start now with some abstract nonsense conceived during my time at the SMALL reu this summer.

See part 2 here.

Basics of \(\ell\)-pieces

I’m going to briefly introduce and motivate a geometric topology construct, called an \(\ell\)-piece, to appear in an upcoming paper from my group in the SMALL reu. I’ll endow these with a notion of hyperbolicity and hyperbolic volume, which also appears in this paper.

In a future post, I’ll briefly introduce the concept of an \(n\)-fold category, guided by the \(n=2\) case of a double category, and show how the collection of \(\ell\)-pieces can be made into the \(\ell\)-cells of an \(\ell\)-fold category. I’ll attempt to describe a convenient \(\ell\)-fold subcategory of such hyperbolic pieces.

Keep in mind that a few of the results are not completely worked out yet. I’ll try to weaken such results accordingly.

What \(\ell\)-pieces are

I’m going to talk intuitively about 3-manifolds; for the purpose of this blog, we may assume that these are either smooth or piecewise-linear, however we will talk purely in the smooth category. All manifolds have boundary. Not all surfaces in a manifold are properly embedded, and they are usually not connected.

Definition. We say the data \((M,\cbr{V_1,V_2})\) are a piece if \(M\) is a 3-manifold and \(V_1,V_2 \subset \partial M\) are spaces satisfying one of the two following conditions:

  • \(V_1\) and \(V_2\) are surfaces and \(V_1 \cap V_2\) is a properly embedded 1-submanifold of \(\partial M\).

  • \(V_1 = V_2 = \varnothing\).

We further say that the data \(\prn{M,\prn{\cbr{V_{2i},V_{2i + 1}} \mid 1 \leq i \leq \ell}}\) are an \(\ell\)-piece if each tuple \((M,\cbr{V_{2i},V_{2i + 1}})\) is a piece and the intersections \(V_i \cap V_j\) are each contained in a 1-manifold.

Remark: this definition actually differs somewhat from the analogous definition in the paper; this definition is a tiny bit more complicated, in a way which does not help with hyperbolic geometry but does help with abstract nonsense.

The intersection condition is necessary for the following construction:

Definition. Suppose \((M,\prn{\cbr{V_i}})\) and \((M',\prn{\cbr{V'_i}})\) are two \(\ell\)-pieces and suppose \(\psi:V_2 \rightarrow V'_1\) is a diffeomorphism. The composition \((M,\prn{\cbr{V_i}}) \circ_\psi (M',\prn{\cbr{V'_i}})\) is an \(\ell\)-piece defined as follows:

$$ \prn{M \cup_\psi M', \left(\cbr{V_1,V'_2},\cbr{V_3 \cup V'_3,V_4 \cup V'_4},\cbr{V_5 \cup V'_5,V_6 \cup V'_6},\dots\right)}. $$

This feels painfully topological, but it gets a little bit easier with some examples. We’ll use examples from knot theory, which require a bit more definitions.

Definition. Suppose \(M\) is a 3-manifold and \(\Sigma \subset \partial M\) is a surface in the boundary of \(M\). Then, We say that a tangle in \((M,\Sigma)\) is a compact smooth properly embedded 1-submanifold \((\mathcal{T},\partial \mathcal{T}) \subset (M,\Sigma)\). If \((M,\prn{\cbr{V_i}})\) is an \(\ell\)-piece, we may define a tangle in \((M,\prn{\cbr{V_i}})\) to be a tanle in \((M,\bigcup_i V_i - \bigcup_{ij} V_i \cap V_j)\). The complement of such a tangle determines a piece. There is an evident notion of composition of tangles, and composition commutes with taking complements.

The following section is not strictly necessary for understanding the \(\ell\)-fold category structure, but it will be nice for motivating this from the geometric topology angle.

Hyperbolic \(\ell\)-pieces.

Note that a 0-piece is simply a 3-manifold. We give a notion of hyperbolicity of \(\ell\)-pieces which bootstraps from the relevant notion for 3-manifolds; we say that a 3-manifold is hyperbolic if it possesses a complete hyperbolic metric of finite volume. In this case, Mostow-Prasad rigidity asserts that this metric is uniquely determined by the homotopy type of the manifold, and in particular the volume of the manifold is a homotopy invariant.

Definition. Suppose \((M,\cbr{V_1,V_2})\) is a piece. Then, we define the \(2m\)-replica to be the 3-manifold

$$ D^{2m}(M,\cbr{V_1,V_2}) := \frac{2n \cdot M}{(V_1,2i) \sim (V_1,2i+1) \hspace{10pt} \text{ and } \hspace{10pt} (V_2,2i-1) \sim (V_2,2i)} $$

where \(2n \cdot M\) is the \(2n\)th copower \(M \coprod M \coprod \cdots \coprod M\). If \((M,\prn{\cbr{V_i}})\) is an \(\ell\)-piece, we define the \(\bm = (2m_1,\dots,2m_\ell)\)-replica to be the 3-manifold

$$ D^{(2m_1,\dots,2m_\ell)}(P,\prn{\cbr{V_i}}) := D^{(2m_1,\dots,2m_{\ell-1})}\prn{D^{2m_\ell}(P,\prn{V_{2\ell - 1},V_{2\ell}}),\prn{D^{m_\ell}(V_1),\dots,D^{m_\ell}(V_{2(\ell - 1)})}} $$

We say that \((M,\prn{\cbr{V_i}})\) is \(\bm\)-hyperbolic if \(D^{\bm}(M,\prn{\cbr{V_i}})\) is hyperbolic, in which case we define the volume

$$ \vol^{\bm}\prn{m,\prn{cbr{v_i}}} := \frac{\vol\prn{D^{\bm}\prn{m,\prn{\cbr{v_i}}}}}{\prod_i 2m_i}. $$

Much of the forthcoming paper is dedicated to a precise form of the following theorem, and a special case of the following conjecture.

Pseudo-theorem. Suppose there exists a suitable collection of \(m\) surfaces decomposing an \(\ell\)-piece into a collection of \((m_1,\dots,m_\ell,m)\)-hyperbolic \((\ell+1)\)-pieces. Then, the \(\ell\)-piece is \((m_1,\dots,m_\ell)\)-hyperbolic, and it’s \((m_1,\dots,m_\ell)\)-hyperbolic volume is lower bounded by the sum of the \((m_1,\dots,m_\ell,m)\)-hyperbolic volumes of the associated \((\ell+1)\)-pieces. In particular, one may find a lower bound on the volume of a 3-manifold by separating it into pieces.

Conjecture. Suppose an \(\ell\)-piece is \((2m_1,\dots,2m_\ell)\)-hyperbolic. Then, it is \((2(m_1 + 1),2m_2,\dots,2m_\ell)\)-hyperbolic.

The conjecture is known for a more specific class, given by the tangles gotten by decomposing links residing in a cycle in \(S^3\). I won’t go into depth here, but we’re looking to prove it in some greater generality. These yield the following corollary:

Corollary. The composition of \(n\)-many \((2m_1n,2m_2,\dots,2m_\ell)\)-hyperbolic $\ell$-pieces is \((2m_1,\dots,2m_\ell)\)-hyprbolic. If the conjecture is true, then the composition of an arbitrary number of \(\bm\)-hyperbolic \(\ell\)-pieces is \(\bm\)-hyprbolic.

This is nice: it appears to tell us we can construct a nice sub-categorical object corresponding with the \(\bm\)-hyperbolic \(\ell\)-pieces.

We’ll work with this in a future blog post; next time, I’ll define \(\ell\)-fold categories and attempt to describe their compositionality with more abstract nonsensical language.