# Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$?

Let $$M$$ be a connected open topological $$d$$-manifold (without boundary).

Whitehead showed that if $$M$$ has a PL structure, there exists a subcomplex of dimension $$\leq d-1$$ onto which $$M$$ deformation retracts.

Can we still find a homotopy equivalent CW complex of dimension $$\leq d-1$$ when $$M$$ is not PL?

• I think this is true. If $X$ is homotopy equivalent to a (connected) CW complex and $H^i(X;M)=0$ for all $\pi_1X$-modules $M$ and all $i>n$, then $X$ is a homotopy equivalent to a CW complex of dimension $\leq n$, provided $n\geq3$. The cases $n=1,2$ can be sorted out by hand (although I only know how to do $n=2$ under some basic finiteness assumptions). Bearing in mind that I am quoting 20+ year old literature, so maybe there is no problem at all. Jul 11, 2020 at 13:25
• @Tyrone Manifolds of dimension $\leq 3$ always have smooth, hence PL structures, so those are certainly okay. Jul 11, 2020 at 13:31
• It is explained in mathoverflow.net/questions/201944/… that a topological $n$-manifold has homotopy type of an $n$-dimensional complex of dimension $\le n$. It remains to exclude dimension $n$ if the manifold is open. Jul 11, 2020 at 13:32
• The proof for $\le n$ should be generalizable to $<n$, i.e. one has to check that the universal covering of your open $n$-manifold has zero cohomology in dimensions $\ge n$ with any local coefficients. Jul 11, 2020 at 13:38
• A universal cover has no nontrivial local systems. I think you mean the cohomology of the actual manifold. It holds by Poincaré duality. Jul 11, 2020 at 22:08

$$\DeclareMathOperator{\co}{H}$$ $$\DeclareMathOperator{\ch}{C}$$ $$\newcommand{\zz}{\mathbb{Z}}$$ $$\newcommand{\nn}{\mathbb{N}}$$ $$\newcommand{\A}{\mathcal{A}}$$ $$\newcommand{\B}{\mathcal{B}}$$ $$\DeclareMathOperator{\lf}{lf}$$
Let me put together an answer following the pointers in the comments. By Whitehead's result stated in the question and smoothability in lower dimensions we may assume $$d \geq 4$$. Write $$\pi := \pi_1(M)$$ for brevity.
As an ANR, M has the homotopy type of a CW-complex, so by a result of Wall it suffices to show that $$\co^{j}(M; \A) = 0$$ whenever $$j \geq d$$ and $$\A$$ is a $$\zz \pi$$-module. Writing $$w$$ for the orientation $$\zz \pi$$-module, by Poincaré duality we have $$\co^j(M;\A) \cong \co^{\lf}_{d-j}(M; \A \otimes_{\zz} w)\,,$$ where $$\co^{\lf}_{*}$$ denotes the locally finite singular homology (sometimes called the Borel-Moore homology). Therefore the only nontrivial thing to check is the vanishing of the 0-th locally finite homology for every $$\zz \pi$$-module $$\B$$. Writing $$\tilde{M}$$ for the universal cover of $$M$$, this amounts to showing that the first differential $$\partial_1 \otimes_{\zz\pi} \B \colon \ch^{\lf}_1(\tilde{M}) \otimes_{\zz\pi} \B \rightarrow \ch^{\lf}_0(\tilde{M}) \otimes_{\zz\pi} \B$$ of the locally finite singular chain complex is surjective, for which $$\partial_1$$ being surjective before tensoring is enough. We can verify $$\partial_1$$ is surjective by elementary means: Fix a locally finite singular 0-chain $$\sigma$$; it is necessarily supported on a discrete subset of $$\tilde{M}$$. Since $$\tilde{M}$$ is non-compact (and second-countable), we can find a countably infinite discrete subset $$\{x_n : n \in \nn\} \subseteq\tilde{M}$$ which contains the support of $$\sigma$$. Thus $$\sigma$$ is a formal sum of the form $$\sigma = \sum_{n \in \nn}a_n x_n$$ with $$a_n \in \zz$$. Now for each $$n \in \nn$$ we can find a path $$\gamma_n : [0,1] \rightarrow M$$ connecting $$x_{n+1}$$ to $$x_{n}$$ such that the formal sum $$\tau := \sum_{n \in \nn}b_n \gamma_n$$ with the coefficients $$b_n := \sum_{j \leq n} a_j$$, is a locally finite $$1$$-chain with $$\partial_1(\tau) = \sigma$$.