© 1996, Topology Atlas

The problem of finding, for each n, the smallest Euclidean space in which the real projective space RP^n (resp. the complex projective space CP^n) can be immersed has attracted the attention of many algebraic topologists during the past 50 years. In [Contemporary Math. 146 (1993)], I wrote a survey article describing the best results and most effective methods that have been obtained on this problem.

In [Annals of Math. 120 (1984)], I proved that if a(m) denotes the number of 1's in the binary expansion of m, then RP^(2(m+a(m)-1)) cannot be immersed in R^(4m-2a(m)). This result, which was obtained by studying the BP-primary obstruction, is close to all known nonimmersion results, and improves on some by arbitrarily large amounts. However, there are still values of n for which the known immersion and nonimmersion dimensions for RP^n differ by arbitrarily large amounts. In [Contemporary Math. 58 (1987)], I discuss these results. The most sweeping immersions of RP^n are those implied by the construction of nonsingular bilinear maps.

Although very little progress has been made on the immersion question for RP^n since my 1984 paper, there is still much to be done. One particularly vexing case is when n is of the form 2^r - 1. In 1963 James obtained a nonimmersion result for this RP^n in roughly R^(2n-2r). In 1967, Gitler and Mahowald described an approach to proving immersions which seemed to show James's nonimmersion was optimal; however, a gap was later found in their proof.

For CP^n, the best results are those of [Crabb, Proc. Royal Soc. Edinburgh 117 (1991)]. This paper uses obstruction theory and K-theory to obtain both immersions and nonimmersions. He shows that CP^n does not immerse in R^(4n - 2a(n) - e), where e is 0 or 1 and is given by a precise formula. Moreover, he shows that CP^n immerses in R^(4n - 2a(n) - e + 1) if a(n) < 8. Removal of this restriction on a(n) would be a major advance in this subject.