Stable phantom maps between classifying spaces of compact Lie groups (Q2716948)

If \(\mathbb X,\mathbb Y\) are the suspension spectra of spaces \(X,Y\), write \(\text{Ph}^d(\mathbb X,\mathbb Y)\) for the group of homotopy classes of (stable) phantom maps of degree \(d\) from \(\mathbb X\) to \(\mathbb Y\). When \(X=BG\), the classifying space of a topological group \(G\), we obtain \(\mathbb X=\mathbb{BG}\). The following is the somewhat surprising main result, which has superficial similarities with the Segal conjecture for finite groups. NEWLINENEWLINENEWLINETheorem. For any two compact Lie groups \(G\) and \(H\), \(\text{Ph}^d(\mathbb{BG},\mathbb{BH})\) is trivial if \(d\) is even, or if \(d\geqslant-1\), or if either group is finite. However, if \(d\) is odd with \(d\geqslant-3\) and if \(G\) and \(H\) have positive rank, then \(\text{Ph}^d(\mathbb{BG},\mathbb{BH})\) is a rational vector space of cardinality \(2^{\aleph_0}\). NEWLINENEWLINENEWLINEThis implies that the graded group \(\text{Ph}^d(\mathbb{BG},\mathbb{BH})\) only depends on whether or not the ranks of \(G\) and \(H\) are zero. Furthermore, \(\text{Ph}^d(\mathbb{BG},\mathbb{BH})\) has no torsion and \(\text{Ph}^d(\mathbb{BG},\mathbb{BH})\cong \text{Ph}^d(\mathbb{BH},\mathbb{BG})\). The proof uses Segal's splitting of \(Q\mathbb{C}P^\infty\) [\textit{G. Segal}, Q. J. Math., Oxf. II. Ser. 24, 1-5 (1973; Zbl 0266.55009)] and operations in \(K\)-theory.