Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact an alternating bihomomorphism) for $(V,+)$ we can form the group extension $\tilde{V}=(V\times\mathbb{R},\cdot)$, where $$(u,\alpha)\cdot(v,\beta)=(u+v,\alpha+\beta-\omega_1(u,v))$$ and $\tilde{V}$ just has the product topology. Similarly we can form an extension $\tilde{W}$ of $W$.
These arise as invariants in my work and I have a (rather tedious) proof of the following crucial result: $$ \tilde{V}\cong \tilde{W}~\Rightarrow~V\cong W \quad (\text{as topological groups}).$$
Questions:
Are there some good references for this or similar results?
Is this result just a consequence of some general cohomological machinery?
Related to the first question:
- Given abelian groups $G,H$, and extensions $\tilde{G},\tilde{H}$ by $K$ when does $\tilde{G}\cong\tilde{H}$ imply $G\cong H$?
My feeling is that this is a small chunk of a more general result, but I don't know enough group theory/cohomolgy to see it. For instance if $G,H$ are torsion-free abelian groups and $\tilde{G},\tilde{H}$ extensions of $G,H$ by alternating bihomomorphisms with values in $\mathbb{Z}$ then an almost identical proof to the TVS case shows that $\tilde{G}\cong\tilde{H}\Rightarrow G\cong H$