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It absolutely doesn’t matter if your sitemap doesn’t include the homepage. Google found and indexed your homepage a long time ago and it will keep crawling it regularly anyway, since you almost certainly have links to it somewhere on your site (like in breadcrumbs, your logo, or elsewhere).
Now here’s the most important part: Google uses sitemaps primarily to discover new pages. Unfortunately, in practice, search bots often don’t re-crawl or update existing pages just because they’re listed in the sitemap even if you’ve specified the proper change frequency or priority rules.
