Binding to static property

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}"/>

You can't bind to a static like that. There's no way for the binding infrastructure to get notified of updates since there's no DependencyObject (or object instance that implement INotifyPropertyChanged) involved.

If that value doesn't change, just ditch the binding and use x:Static directly inside the Text property. Define app below to be the namespace (and assembly) location of the VersionManager class.

<TextBox Text="{x:Static app:VersionManager.FilterString}" />

If the value does change, I'd suggest creating a singleton to contain the value and bind to that.

An example of the singleton:

public class VersionManager : DependencyObject {
    public static readonly DependencyProperty FilterStringProperty =
        DependencyProperty.Register( "FilterString", typeof( string ),
        typeof( VersionManager ), new UIPropertyMetadata( "no version!" ) );
    public string FilterString {
        get { return (string) GetValue( FilterStringProperty ); }
        set { SetValue( FilterStringProperty, value ); }
    }

    public static VersionManager Instance { get; private set; }

    static VersionManager() {
        Instance = new VersionManager();
    }
}

<TextBox Text="{Binding Source={x:Static local:VersionManager.Instance},
                        Path=FilterString}"/>