This study will evaluate the efficacy, safety, and pharmacokinetics of adjuvant atezolizumabwhen given in combination with trastuzumab emtansine compared with placebo andtrastuzumab emtansine for patients with residual invasive HER2-positive breast cancerfollowing neoadjuvant taxane-based and HER2-targeted therapy including trastuzumab, whoare at high risk of disease recurrence. The primary endpoint for this study is invasivedisease-free survival (IDFS).

To compare how often cancer recurs (comes back) after 3 weeks of radiation compared to 5 weeks of radiation in patients who receive radiation therapy delivered to the lymph nodes near the breast. The side effects that can develop during or after radiation treatment, including how often arm swelling (edema) happens, will also be studied

RATIONALE: Estrogen can cause the growth of breast cancer cells. Hormone therapy using tamoxifen citrate, goserelin acetate, leuprolide acetate, anastrozole, letrozole, or exemestane, may fight breast cancer by lowering the amount of estrogen the body makes. Everolimus may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet know whether hormone therapy is more effective when given with or without everolimus in treating breast cancer. PURPOSE: This randomized phase III trial studies how well giving hormone therapy together with or without everolimus work in treating patients with breast cancer.