The World Health Organization stated its suspicions that aluminum links up to Alzheimer's way back in 1993. Used long-term, "it might be responsible for the tangles and plaques in the brain of Alzheimer patients." It most certainly has been shown to accumulate in the brain and is classified as a "heavy metal" (due to specific gravity)(2).
Aluminum isn't the only antiperspirant or regular deodorant danger, though. Commonly-used preservatives BHT or parabens mime estrogen and thus disrupt the body. Formaldehyde isn't just used for corpses by far: and it causes cancer. Steareths are also carcinogens (cancer-causes). Triclosan, the once more widely-touted antibacterial agent, is an endocrine (hormone) disrupter. Phthalates and musks are further endocrine disrupters: the trouble is, laws have protected companies from needing to identify different "fragrances," which could be these and potential hidden allergens which certain sensitive individuals couldn't find directly on the label(1). A middle-aged man got severe muscle pain from antiperspirants(2)!
So, the alternative is to keep what comes out of the armpits from reeking in a natural way with pure, natural ingredients(2). [I shower with natural soap and wear natural deodorant. Some natural deodorants especially work for different people better than others, from my experience. Some stores carry mineral salt sticks - I've used that, a coconut deodorant, and a sage herbal stick and spray.]
Sources:
1. Chia, Jessica. "The Crazy Ways Antiperspirant Messes With Your Health." Prevention, Hearst Magazine Media, Inc. 11 November 2013. Web. 14 April 2020. https://www.prevention.com/life/g20496231/the-crazy-ways-antiperspirant-messes-with-your-health/?slide=1.
2. "Deodorants and antiperspirants side effects." Vaughter Wellness, Gromako Data & Services Ltd. https://owndoc.com/side_effects/cosmetics/deodorants-antiperspirants/.