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Projective elements in \(K\)-theory and self maps of \(\Sigma CP^{\infty}\). (English) Zbl 0924.55002
The author works in the homotopy category of based spaces and based maps. Given a space \(X\), let the reduced \(K\)-theory be denoted by \(K(X)\) and the homology group of integral coefficients by \(H_*(X)\). Let \(\mathbb{C}\text{P}^\infty\) be the infinite-dimensional complex projective space. Let \(\eta\) be the canonical line bundle over \(\mathbb{C}\text{P}^\infty\) and \(i:\mathbb{C}\text{P}^\infty\to \text{BU}\) be the classifying map of the virtual bundle \(\eta-1\). Since BU has a loop space structure, there exists a unique extension of \(i\) to the loop map \(j:\Omega\Sigma \mathbb{C}\text{P}^\infty\to \text{BU}\).
In this paper, the author investigates the following problems: Given an element \(\alpha\in K(X)\), when does there exist a lift \(\widehat\alpha\in [X,\Omega\Sigma \mathbb{C}\text{P}^\infty]\) such that \(j_*(\widehat\alpha)= \alpha\)? If \(\alpha\) has a lift, how can we construct the lift \(\widehat\alpha\)? Define \(\text{PK}(X)= \{\alpha\in K(X)\mid\exists\widehat\alpha\in [X,\Omega\Sigma \mathbb{C}\text{P}^\infty]\) such that \(j_*(\widehat\alpha)= \alpha\}\). If an element \(\alpha\in K(X)\) belongs to \(\text{PK}(X)\), one says that \(\alpha\) is projective. The significance of the above problem is as follows: The James splitting theorem [I. M. James, “The topology of Stiefel manifolds”, Lect. Note Series 24 (1976; Zbl 0337.55017)] implies that there exists a loop map \(\theta: \text{BU}\to \Omega^\infty \Sigma^\infty\mathbb{C}\text{P}^\infty\) such that \(\theta\circ j= E^\infty: \Omega\Sigma\mathbb{C}\text{P}^\infty\to \Omega^\infty\Sigma^\infty \mathbb{C}\text{P}^\infty\). Therefore, given an element \(\alpha\in K(X)\), the stable map, \(\text{ad }j\circ(\theta(\alpha)): \Sigma^\infty X\to\Sigma^\infty \mathbb{C}\text{P}^\infty\) can be considered. Using the information of \(K(X)\), the induced homomorphism [C. A. McGibbon, Trans. Am. Math. Soc. 271, 325-346 (1982; Zbl 0491.55014); K. Morisugi, Publ. Res. Inst. Math. Sci. 24, No. 2, 301-309 (1988; Zbl 0657.55010)] of \(\text{ad }j \circ(\theta(\alpha))_*: H_*(X)\to H_*(\mathbb{C}\text{P}^\infty)\) can be calculated. If \(\alpha\) has a lift \(\widehat\alpha\), then this implies that the stable map \(\text{ad }j\circ(\theta(\alpha))\) and its induced homomorphism come from the unstable map \(\text{ad }j\circ(\widehat\alpha): \Sigma X\to \Sigma\mathbb{C}\text{P}^\infty\). These imply that the determination of \(\text{PK}(X)\) gives complete information of the image of the homomorphism: \([\Sigma X,\Sigma\mathbb{C}\text{P}^\infty]\to \operatorname{Hom}(H_*(X), H_*(\mathbb{C}\text{P}^\infty))\). However, since the above factors through \(\operatorname{Hom}(H_*(X), H_*(\Omega \Sigma\mathbb{C}\text{P}^\infty))\), it is desirable to obtain the image of \([X,\Omega\Sigma \mathbb{C}\text{P}^\infty]\to \operatorname{Hom}(H_*(X), H_*(\Omega\Sigma \mathbb{C}\text{P}^\infty))\). So, if possible, it is preferable to have the information not of \(\text{ad }j\circ(\widehat\alpha)_*\) but \(\widehat\alpha_*: H_*(X)\to H_*(\Omega\Sigma \mathbb{C}\text{P}^\infty)\). For this the geometry of the lift \(\widehat\alpha\) is necessary.
In this context, the author proves five theorems and a corollary. In Theorem 1.1 it is proved that if \(\mathbb{C}\text{P}^\infty\wedge \mathbb{C}\text{P}^\infty\to \mathbb{C}\text{P}^\infty\) is the adjoint of the Hopf construction for the H-space \(\mathbb{C}\text{P}^\infty\), then this map has an extension \(\#:\Omega\Sigma \mathbb{C}\text{P}^\infty\wedge \Omega\Sigma\mathbb{C}\text{P}^\infty\) such that \(j\circ\#= \bigotimes\circ(j\wedge j)\), where \(\bigotimes: \text{BU}\wedge \text{BU}\to \text{BU}\) is the map which represents the external tensor product \(K(X)\otimes K(Y)\to K(X\wedge Y)\). In Theorem 1.2 the properties of \(\text{PK}(X)\) are established. Theorem 1.4 contains an evaluation of the commutator in the group \([\mathbb{C}\text{P}^\infty, \Omega\Sigma\mathbb{C}\text{P}^\infty]\) and in the Theorem 1.5 the Hurewicz homomorphism \(h: \pi_*(\Omega\Sigma \mathbb{C}\text{P}^\infty)\to H_*(\Omega\Sigma \mathbb{C}\text{P}^\infty)\) is studied. As a corollary of Theorem 1.5 it is proved that the group \([\Sigma \mathbb{C}\text{P}^n,\Sigma \mathbb{C}\text{P}^n]\) is not commutative for \(n\geq 3\). Theorem 1.7 is a technical result concerning the composition structure of the adjoints of some maps \(f_n: \mathbb{C}\text{P}^\infty\to \mathbb{C}\text{P}^\infty\) inductively defined starting from the inclusion \(f_1: \mathbb{C}\text{P}^\infty\to \mathbb{C}\text{P}^\infty\).
Reviewer: Ioan Pop (Iaşi)

MSC:
55N15 Topological \(K\)-theory
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