# A variant of $\ell^2$-cochains

Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some number that depends on $k$. Let $X_k$ be the set of $k$-cells.

Let $\ell^2_k(X)$ be the set of functions $a : X_k \to \Bbb R$, such that the series $$\sum_{e \in X_k} a(e)^2$$ converges (this implicitly makes use of the counting measure on $X_k$). Then the incidence bound assumption implies that coboundary operator $$\delta: \ell^2_k(X) \to \ell^2_{k+1}(X)$$ is defined (this uses the same formula that arises when defining the cellular cochain complex of $X$).

When $\dim X =1$ this construction was introduced by Dodziuk and Kendall in

Dodziuk, J.(1-CUNYG); Kendall, W. S.(4-STRA) Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), 68–74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986.

Questions

1. Has this construction been investigated in the generality described above?

2. How is the cohomology of this complex related to the usual cellular cohomology of $X$?

3. Is there a set of reasonable conditions on $X$ which guarantee that this cohomology is finite dimensional?

4. How does the above relate to other notions of $L^2$-cohomology?

I'm not sure if this should be a comment rather than an answer, since I'm not attempting to answer the hard questions (2) and (3). But it's too long to be a comment. That said:

First of all, I just want to clarify that presumably you want incidence to be bounded with multiplicity, that is, the coefficients of the boundary map to also be bounded.

For (4), take a look at this technical report by Attie and Block: https://pdfs.semanticscholar.org/cfd5/297840f262e255f86777e112757dbffdf9e5.pdf

Theorem 4 is a de Rham theorem for $L^p$ cohomology -- for a manifold with bounded geometry, $L^p$ cohomology is equal to simplicial $L^p$ cohomology with real coefficients, on any given bounded triangulation. (I suppose you can also define a "piecewise de Rham" cohomology and then the isomorphism holds for any simplicial complex, in standard and $L^p$ flavors.)

Now if you have a manifold with a nice enough CW structure, then you could replace each cell with a bounded number of simplices up to homotopy equivalence, and you would have an isomorphism from your theory to the "usual" de Rham version. The pullback to the universal cover of a CW structure on a compact manifold is of course nice enough, but a CW complex of bounded incidence is not. E.g. take an $\mathbb{R}$ and at the $n$th lattice point attach an $S^2$ which in turn has a 4-cell attached to it via a map with Hopf invariant $n$. This has bounded incidence, since only the 1-cells are incident to anything at all, but an infinite number of homotopy types of attaching maps and so a simplicial complex version would have to get more and more complicated as you go further out.

I don't know of anyone studying the $L^2$ version in this level of generality. I studied the $L^\infty$ analogue at some length in my thesis (see sections 5 and 6 of https://arxiv.org/abs/1410.3368), but I was only concerned with universal covers of compact spaces, which avoids a lot of technical issues.

• " presumably you want incidence to be bounded with multiplicity, that is, the coefficients of the boundary map to also be bounded." Yes, I believe I do. – John Klein Aug 14 '17 at 13:57