Is there a link to hompage that purges the cache – problem with ajax login wiget

I have a website a ajax login widget. The problem is, on te homepage only, it doesn’t update when a user logs in, only if the user refreshes the page. I can manually a redirect link when a user logs in, so is it possibe to direct them to a homepage link that purges te cache?

https://www.ads-software.com/plugins/w3-total-cache/