# zbMATH — the first resource for mathematics

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
##### Keywords:
projective element; lift
Full Text: