Real Projective Space

Abstract

The objective of this chapter is to present the cohomological calculations (in MU*, KU* and -BP*) which will be needed in this and later chapters for the study of maps of the form

$$ g:\sum ^\infty S^{2^{k + 1} - 2} \to \sum ^\infty \mathbb{R}\mathbb{P}^{2^{k + 1} - 2} $$

and related Whitehead product maps. In [30] it is shown that if g* is non-zero on jo*-theory, which was introduced in Chapter 1, Example 1.3.4(iv), then g is detected by Sq^2k. On the other hand detection by jo*-theory is equivalent to the KU*-e-invariant (defined by means of ψ³-1) being $\frac{{(3^{2^k } - 1)(2w + 1)}} {4}$. The calculations of this chapter will eventually be used to prove both this and the converse result (conjectured in [30]) in Chapter 8, Theorem 8.1.2.