Maps to Spaces in the Genus of Infinite Quaternionic Projective Space

Abstract

Abstract. Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An interpretation of these results in the context of Adams-Wilkerson embedding in integral K-theory is also given.

Representation Homology of Topological Spaces

“…Even though the spaces B ≃ BG do not admit maximal tori, this does not rule out the possibility that there could exist interesting maps f : BT → B whose homotopy fibres are not finite CW complexes. In his thesis (see [Yau04]), D. Yau refined Rector's classification by describing the spaces B in the genus of BSU (2) that can occur as targets of essential (i.e., non-nullhomotopic) maps from BT . Such spaces admit a beautiful arithmetic characterization: Theorem 4.5 (Yau).…”

“…Next, recall that, by Theorem 4.5, among all essential maps BT → B, there is a 'maximal' one p B : BT → B, for which deg (p B ) = N B , where N B is the integer defined by (4.3): the corresponding power series (5.14) p * B (u) = N B t 2 + higher order terms in t . is a useful K-theoretic invariant of B that depends on the Rector invariants (B/p) (see [Yau04]). Using (5.14), we define a sequence of subrings Q m (B) in Z[[t]] inductively by the rule:…”

“…] can be taken in the form t = bξ, where ξ ∈ F 2 K 2 (BT ) and b ∈ K −2 (pt) is the Bott element (see [Yau04,Sect. 3]).…”