If the binding needs to be two-way, you must supply a path.
There’s a trick to do two-way binding on a static property, provided the class is not static : declare a dummy instance of the class in the resources, and use it as the source of the binding.
<Window.Resources>
<local:VersionManager x:Key="versionManager"/>
</Window.Resources>
...
<TextBox Text="{Binding Source={StaticResource versionManager}, Path=FilterString}"/>