Collared boundary of a non-metrizable manifold

Question

For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally flat imbeddings of topological manifolds", Morton Brown, 1962. Let $M$ be a non-metrizable manifold-with-boundary. Does $M$ have a collared boundary?

Answer 1

A nice recent reference for questions about non-metrisable manifolds is David Gauld's book aptly named "non-metrisable manifolds". For instance it is shown that any metrisable component of the boundary of a manifold (metrisable or not) is collared (Corollary 3.11 on page 44, it is an almost immediate consequence of R. Connely proof of M. Brown's result).

This is not true if the component is non-metrisable: Example 1.29 on p. 16 (which is originally due to P. Nyikos) gives a description of a manifold whose interior is $\mathbb{R}^2$ and whose boundary is the open long ray $\mathbb{L}_{+}$. This boundary component cannot be collared because there is no embedding sending $\mathbb{L}_{+}\times[0,1]$ into the manifold.

It is not clear to me whether the usual definition of `collared' is only for connected components of the boundary or if there is a global one asking for an embedding from $\partial M\times[0,1]$ into the manifold $M$. In that case a simpler counter example is the Prüfer surface (dating back to Rado), which is also detailed in D. Gauld's book in Example 1.25.

EDIT: Comments by the OP (and silly ones by myself) made me realize that something was a bit unclear in the statement of Theorem 3.10 in Gauld's book, from which Corollary 3.11 follows. Here are some details.

Theorem 3.10 states that if a closed subset $B$ of a Hausdorff space $X$ is locally collared and strongly paracompact, then $B$ is collared. But actually, what is needed is a bit stronger: that $B$ is strongly paracompact in $X$, that is, given a cover $\mathcal{U}$ of $B$ by open sets of $X$, there is another cover $\mathcal{V}$ of $B$ by open sets of $X$ such that each member of $\mathcal{V}$ is contained in a member of $\mathcal{U}$ and $\mathcal{V}$ is star-finite in $X$. (Star-finite means each member intersects finitely many members.) I actually asked D. Gauld about it, and he agreed that he has been a little bit careless in the statement.

This requirement is stronger, since for instance any manifold whose components are metrisable is strongly paracompact. The boundary $\partial P$ of the Prüfer surface $P$ is a discrete union of continuum many real lines and hence is strongly paracompact (in itself, so to say). But $\partial P$ is not strongly paracompact in $P$, and actually $\partial P$ is not collared in $P$.

A consequence of Theorem 3.10 (amended) is that if the boundary $\partial M$ of a manifold $M$ is made of (at most) countably many metrisable components, then it is collared. The proof is by using the fact that Lindelöfness is equivalent to metrisability and strong paracompactness for connected manifolds to cover $\partial M$ by countably many euclidean sets whose union yields a strongly paracompact submanifold of $M$ also having $\partial M$ as a boundary.

Answer 2

Mathieu Baillif provided an answer to the question. I'll leave here some more notes for future readers. The following strengthening of the theorem in Gauld's book "Non-metrizable manifolds" holds:

Let $X$ be a manifold-with-boundary. Then $\partial X$ is collared in $X$ and $\partial X$ is paracompact in $\partial X$ if and only if $\partial X$ is paracompact in $X$.

This is quite a nice theorem. However, it leaves open the characterization of of non-paracompact collared boundaries. I would be interested to hear if someone can complete the characterization.

The following example shows that there exists non-paracompact collared boundaries. The open long ray is a non-paracompact manifold. Therefore, its cartesian product with the interval $[0, 1]$ is a manifold-with-boundary whose boundary is non-paracompact and clearly collared.

Some background

An $X$-collar of $X' \subset X$ is a function $h : X' \times [0, 1] \to X$, such that

• $h$ is an embedding,
• $h(x, 0) = x$, for each $x \in X'$,
• $h(X' \times [0, 1))$ is open in $X$

A subset $X' \subset X$ is $X$-collared, if there exists an $X$-collar of $X'$.

A local collar of $X' \subset X$ in $X$ is a pair $(U, h)$, where $U \subset X'$ is open in $X'$, and $h : \overline{U}(X') \times [0, 1] \to X$ is an embedding such that

• $h^{-1}(X') = \overline{U}(X') \times \{0\}$,
• $h(x, 0) = x$, for each $x \in \overline{U}(X')$,
• $h(U \times [0, 1))$ is open in $X$.

A subset $X' \subset X$ is locally collared in $X$, if for each $x \in X'$ there exists a local collar $(U, h)$ of $X'$ in $X$ such that $x \in U$.

Here are some properties of collared subsets, which I think to have proved:

• If $X' \subset X$ is $X$-collared, then $X' \times Y$ is $(X \times Y)$-collared. Applied to manifolds-with-boundary, if $X$ is a manifold-with-boundary, $Y$ is a (boundaryless) manifold, and $\partial X$ is $X$-collared, then $X \times Y$ is a manifold-with-boundary, and $\partial (X \times Y) = \partial X \times Y$ is $(X \times Y)$-collared.
• If $X'_i \subset X_i$ is $X_i$-collared for each $i \in I$, then $\sqcup X'_I$ is $\sqcup X_I$-collared, where $\sqcup$ denotes disjoint sum.
• Let $X' \subset X$, $\mathcal{U} \subset \mathcal{P}(X')$ be an $X'$-open cover of $X'$, $V_U = \overline{U}(X') \times [0, 1]$ for each $U \in \mathcal{U}$, and $h : X' \times [0, 1] \to X$. Then $h$ is a collar of $X'$ in $X$ and $\mathcal{U}$ is $X'$-locally finite if and only if $h$ is injective, $(U, h|V_U)$ is a local collar of $X'$ in $X$ for each $U \in \mathcal{U}$, and $\{h(V_U) : U \in \mathcal{U}\}$ is $X$-locally finite.