We know that we can put two different structures on $\mathbb R^5$ in the topological category. First is $C(S^{4})$ and second is $C(\Sigma X^3)$, where $X^3$ is a homology sphere and $C(\cdot)$ stands for the open cone construction and $\Sigma(\cdot)$ stands for the suspension construction.


The question is: do other cone structures exist? i.e. if $C(Y^{n-1})$ is homeomorphic to $\mathbb R^n$, is it true that $Y$ is $(n-1)$-sphere or the suspension of a $(n-2)$-homology sphere?



Cone structures on $\mathbb R^n$
You should look at this paper:
<http://www.ams.org/journals/bull/1978-84-05/S0002-9904-1978-14527-3/S0002-9904-1978-14527-3.pdf>


and in particular theorem 3.5. 


And also 
<http://www.maths.ed.ac.uk/~aar/books/haupt.pdf>
p. 23. 


It follows from the double suspension theorem and its corollaries that when $n\geq 5$ you can take for $Y^{n-1}$ any combinatorial homology manifold wich is homotopy equivalent to $S^{n-1}$, then the cone $C(Y^{n-1})$ is homeomorphic to $\mathbb{R}^n$.


Take for example two homology $3$-spheres $Z$ and $Z'$ consider $Z\times [0,1]$ and $Z'\times [0,1]$, build the connected sum $W$ (just remove a $4$-disk inside $Z\times ]0,1[$ and $Z'\times ]0,1[$ and glue). $W$ is a manifold with $4$ boundary components, glue a cone on each of this boundary component, you get a space $Y$ with $4$ isolated singularities whose links are homology spheres. Thus $Y$ is a combinatorial homology manifold, it is $1$-connected (by Van-Kampen theorem) and has the homology of a $4$-sphere (by Mayer-Vietoris theorem). And take the cone over $Y$.


