With W3 Total Cache when I publish a post it does not appear in the homepage. Only if i purge all cache [closed]

You are most likely using browser caching ? Therefore new visitors would see the newly published post just fine, but because you’ve previously visited the page its cached and therefore your browser cache hasnt expired.

Generally its good practice to clear all caches upon publishing posts/pages and/or activating or deactivating a plugin.

You can also disable page caching for the front page :

performance - page cache -  Don't cache front page

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