A colleague asked me this question recently. Every injective continuous map between manifolds of the same (finite) dimension is open - this is Brouwer's Domain Invariance Theorem. Is the same true for complete boundaryless Alexandrov spaces (of curvature bounded below)?


Alexandrov spaces are manifolds almost everywhere, and their singularities have special structure. In dimensions 1 and 2 there are no topological singularities (all Alexandrov spaces are manifolds). Higher dimensional singularities have a sort of inductive description: every point $x$ in an $n$-dimensional Alexandrov space has a neighborhood homeomorphic to the cone over an $(n-1)$-dimensional connected(!) Alexandrov space $\Sigma\_x$ which carries a metric of curvature $\ge 1$. The last property implies that $\Sigma\_x$ is compact and its fundamental group is finite.


For example, in dimension 3 the only type of singularity is the cone over $RP^2$. In dimension 4 there are cones over lens spaces, $\mathbb R\times Cone(RP^2)$ and maybe other beasts.


There is a similar purely topological question: is Domain Invariance Theorem true for "almost manifolds"? An "almost manifold" is a pseudo-manifold obtained from $n$-simlices by gluing their $(n-1)$-dimensional faces in pairs - that is, there are exactly two $n$-simplices adjacent to each $(n-1)$-simplex, and there are no extra identifications between lower-dimensional faces. I'm not sure that all Alexandrov spaces admit triangulations, but if they do, they are "almost manifolds" (of a special kind).



Is there Domain Invariance for Alexandrov spaces?
The following lemma from *[Grove--Petersen, A radius sphere theorem](http://www.math.psu.edu/petrunin/papers/alexandrov/radius_sphere_theorem+.pdf)* does the trick.



> 
> **Lemma 1.** Let $X$ be a compact Alexandrov space without boundary. Then $X$ has a fundamental class in Alexander-Spanier cohomology with $\mathbb Z\_2$ coefficients;
>  i.e. $\bar H^n(X,\mathbb Z\_2 ) = \mathbb Z\_2$.
> 
> 
> 




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**Why:** First note that it is true for compact spaces --- in this case the map has $\mathbb Z\_2$-degree one. Moreover, in this case the same it true for any continuous map which is injective around one point in the target.


Now let $X$ and $Y$ be $m$-dimensional Alexandrov spaces,
$\Omega\subset X$ be an open subset and $f:\Omega\to Y$ be an injective continuous map and $y=f(x)$.
One can use $f$ to construct a continuous map between sperical suspensions over spaces of directions $\mathbb S\,\Sigma\_x\to \mathbb S\,\Sigma\_y$ which is injective around one point in the target --- take a small sherical neghborhood $W\ni y$ and collapse everything outside of $W$ to the south pole of $\mathbb S\,\Sigma\_y$.
(We can do it sinse small spherical neighborhood of a point $x$ in an Alexandrov space is homeomorphic to cone over space of directions at $x$.)




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The same is true for the second question --- link of any pseudomanifold is a pseidomanifold, thus it has $\mathbb Z\_2$-fundamental class.


