No evidence links breast cancer to sweating, not sweating, or antiperspirants.
Breast cancer predates antiperspirants by many centuries.
Furthermore, there's no ingredient in antiperspirants known to cause cancer. Fortunately, there's plenty of information on the Internet refuting this claim, including a statement by the American Cancer Society. And it's worth noticing that websites spreading the word about the alleged link between antiperspirants and breast cancer are often selling alternative products to quell underarm odor. Tampons do not contain any asbestos. And TSS, which is very rare today, is not promoted by the fiber content, but by super-absorbent brands.