This is a cross-post of [this question from MSE](https://math.stackexchange.com/questions/4617947/serre-fibrations-between-spaces-of-embeddings-reference-request).




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Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ consisting of the (topological) embeddings $M\to N$. Here $\operatorname{Map}(M,N)$ is equipped with the compact-open topology.
I am looking for a reference of the following assertion (if it is true):



> 
> Let $M$ be a topological manifold of dimension $k$. The evaluation map $$\operatorname{Emb}(\mathbb{R}^k,M)\to M$$ at the origin $0\in\mathbb{R}^k$
> is a Serre fibration (or a Hurewicz fibration, preferably).
> 
> 
> 


Thanks in advance.



Reference request - Fibrations between spaces of embeddings
It is a Serre fibration, and this is one of the rare cases where you can give an elementary argument and don't need something hard like [Edwards-Kirby](https://www.jstor.org/stable/1970753?origin=crossref), [Lees](https://www.projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-3/Immersions-and-surgeries-of-topological-manifolds/bams/1183530549.full), or [Lashof](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-20/issue-1/Embedding-spaces/10.1215/ijm/1256050168.full). I do not know a reference so let me outline the argument.


We construct a lift in a commutative square
$$\require{AMScd}
\begin{CD}
I^j \times \{0\} @>{f}>> \mathrm{Emb}(\mathbb{R}^k,M) \\
@VVV @V{\mathrm{ev}\_0}VV \\
I^j \times I @>{F}>> M
\end{CD}$$
by finding a continuous map $H \colon I^j \times I \to \mathrm{Homeo}\_c(M)$ such that $H(x,0) = \mathrm{id}\_M$ and $H(x,t)(F(x,0)) = F(x,t)$. Then the desired lift will be given by $(x,t) \mapsto H(x,t) \circ f(x)$.


Subdividing the domain $I^j \times I$ of $F$ into fine enough $(j+1)$-cubes such that $F$ sends each of them into a chart of $M$ homeomorphic to $\mathbb{R}^k$, we will construct $H$ inductively over these cubes and thus may assume that $M = \mathbb{R}^k$. We can then define $H(x,t)$ as given by translation by $F(x,t)-F(x,0) \in \mathbb{R}^k$, cut off by a suitable bump function.


