Embeddings of Real Projective Spaces

The question of finding the smallest Euclidean space in which real projective space P n can be (differentiably) embedded was the subject of intense investigation during the 1960s and 1970s. The purpose of this paper is to survey the status of the question, and add a little bit to our knowledge by proving one new family of embeddings, using old methods of obstruction theory. Our new result is given in the following theorem. Theorem 1.1. If n = 2 + 3 ≥ 11, then P n can be embedded in R2n−4.