Toolbar/topbar missing on homepage only?

As Mayeenul wrote: Use Firebug/Chrome Inspector to see what’s wrong.

Look for Javascript Errors in the console see if the Adminbar is inside the DOM but hidden via CSS.

It could also be disabled via a filter in PHP with something like this:

if( is_front_page() ) {
    add_filter('show_admin_bar', '__return_false');
}

… but I would say it’s more likely that you have an error somewhere in the DOM or in your CSS/JS for the frontpage that is hiding the adminbar.